DE112010001894B4 - Method for measuring a surface microstructure, method for data analysis of surface microstructure measurement and surface microstructure measuring system - Google Patents

Abstract

A surface microstructure measuring method for measuring a microstructure (149b) of a sample surface (141, 146), the method comprising the steps of:irradiating the sample surface (141,146) with X-rays at a grazing incidence angle and measuring a scattering intensity;assuming a sample model having a microstructure (149b) on a surface (141, 146) in which one or more layers are formed in a direction perpendicular to the surface (141, 146) and unit structures are periodically arranged in a direction parallel to the surface (141, 146) within the layers , calculating a scattering intensity of X-rays scattered by the microstructure (149b) based on the refractive indices of the respective layers of the one or more layers in a direction parallel to the surface (141, 146) within the layers, and performing a fit of the scattering intensity of the X-ray radiation calculated from the sample model to the measured scattering intensity; anddetermining, as a result of the fitting, an optimal value of a parameter for specifying a shape of the unit structures, in which, assuming that the unit structures have fluctuations in positions relative to an exact periodic position and the fluctuations in positions are only from a relative positional relation between depend on the unit structures, the scattering intensity of the X-rays is calculated in which the unit structures are formed by a uniform substantial area and a uniform void area in the layers, and the scattering intensity of the X-rays caused by the uniform substantial area is calculated.

Description

technical field

The present invention relates to a surface microstructure measurement method, a surface microstructure measurement data analysis method, and a surface microstructure measurement system

State of the art

In a semiconductor manufacturing process, transistors in an LSI are often formed by employing a line and space structure. 30 FIG. 9 is a plan view showing an example of a semiconductor substrate 900. FIG. The semiconductor substrate 900 has a line and space structure made up of line portions 910 and space portions 920; Transistor structures such as gate electrodes are fabricated into such a line and space structure. A gate length is an important parameter for determining the characteristics of a transistor, and suppressing deviations thereof to a certain value or less is a very important factor for determining the performance of an LSI. The minimum length in such an element structure is called the CD (critical dimension). When transistors having greatly different characteristics such as a threshold voltage and a gate current are installed in a circuit, it is possible to ensure the performance of an LSI that is an aggregation thereof. In order to avoid the above problem, it is necessary to constantly optimize the conditions of the semiconductor manufacturing process.

Even in other processes, factors that change the CD of comparable elements are always present. Consequently, in a wafer to be actually manufactured, it is a very important point to check the CD value in due time. Conventionally, for measuring the CD of a semiconductor or the like, a CD-SEM or a light scattering (scatterometry) analysis method is used.

In contrast, a technology is proposed for analyzing the density fluctuation in a non-uniform-density multilayer film where one or more non-uniform-density films are stacked on a substrate by using a scattering function which characterizes an X-ray scattering curve according to a parameter representing the distribution state of a particulate material (e.g. Patent Document 1).

In the analysis method of the non-uniform density multilayer film disclosed in Patent Document 1, the scattering function representing the X-ray scattering curve according to the parameter indicating the distribution state of the particulate matter is used. An X-ray scattering intensity is calculated under the same conditions as the measurement conditions under which the X-ray scattering intensity is actually measured, and a fit is performed between the calculated X-ray scattering intensity while a parameter is changed and the actually measured X-ray scattering intensity, and the parameter value when the calculated X-ray scattering intensity coincides with the actually measured X-ray scattering intensity when determining the distribution state of the particulate matter in the non-uniform density multilayer film. As described above, by using a function, as a scattering function, which introduces a transition probability in which an exact solution of the multilayer film without scattering in an interface is set as an initial state and a final state, the distribution state of the particulate matter of non-uniform density is analyzed.

In a method disclosed in Patent Document 2, for measuring the critical dimension (CD), the surface of a substrate is irradiated with X-rays so as to hit a portion of a periodic structure on the surface of a sample. Then, to measure the dimension of the structure parallel to the surface of the sample, an X-ray pattern resulting from the scattering corresponding to the feature of the surface is detected as a function of the azimuth (Azmuth) parallel to the surface of the sample. However, as a method for measuring the diffraction line of each order of the formed periodic structure, the rotation of the sample itself in the azimuth direction is not explicitly specified.

In contrast, Non-Patent Document 1, which was disclosed prior to the disclosure of Patent Document 1 described above, discloses that a high-resolution measurement system is constructed by using a mirror, a crystal collimator, and an analyzer, wherein a diffracted X-ray of a periodic structure that is formed on the surface as a plane roentgen small angle gene scattering pattern and is measured as a function of an azimuth. In the method disclosed here, the periodic structure is regarded as almost crystal-like, with a sample rotated in the azimuth direction so that the spectrum of each order satisfies the well-known Bragg diffraction condition (in non-patent document 1, this is denoted as ϕ), a very large number of scattering peaks are detected and, based on this, the structure spacing and the line width of the periodic structure are determined with high accuracy. Further, a method is disclosed in which when the sample is rotated in the azimuth direction, a detector is simultaneously rotated at a speed twice that of this (in non-patent document 1, this is called 2θ/φ scan) and thus the Bragg diffraction conditions are met.

• Patentdokument 1: JP 2003 - 202 305 A
• Patentdokument 2: US 2006 / 0 133 570 A1
• Nicht-Patentdokument 1: Yoshiyasu ITO, Katsuhiko INABA, Kazuhiko OMOTE, Yasuo WADA, Tomokazu EZURA, Ken TSUTSUI und Susumu IKEDA, „Evaluation of a Microfabricated Structure by an Ultra-high Resolution In-plane X-ray Small Angle Scattering Method“, The 53th Applied Physics Related Discussion Meeting Preprint 24a-B-4/III, May 24, 2006, No. 3, p. 1471
• Nicht-Patentdokument 2: Yoshiyasu ITO, Katsuhiko INABA, Kazuhiko OMOTE, Yasuo WADA und Susumu IKEDA, Characterization of Submicron-scale Periodic Grooves by Grazing Incidence Ultra-small-angle X-ray Scattering, Japanese Journal of Applied Physics, Japan, The Japan Society of Applied Physics, August 10, 2007, Vol. 46, No. 32, 2007 pp. L773-L775 Die US 2006/0133570 A1 offenbart Merkmale, die unter den Oberbegriff des Anspruches 1 fallen.

Further, Non-patent Document 2 proposes a structural model for theoretically calculating an X-ray scattering spectrum that is a function with respect to the direction of the measured azimuth (equation 2) in Non-patent Document 2. A method is disclosed in which, based thereon , an X-ray scattering intensity is specifically calculated with a parameter being optimized by comparison with the measured X-ray scattering spectrum, whereby a microstructure such as a line width and a slope of a sidewall is determined.

• Patent Document 1: JP 2003 - 202 305 A
• Patent Document 2: U.S. 2006/0 133 570 A1
• Non-patent document 1: Yoshiyasu ITO, Katsuhiko INABA, Kazuhiko OMOTE, Yasuo WADA, Tomokazu EZURA, Ken TSUTSUI and Susumu IKEDA, "Evaluation of a Microfabricated Structure by an Ultra-high Resolution In-plane X-ray Small Angle Scattering Method", The 53th Applied Physics Related Discussion Meeting Preprint 24a-B-4/III, May 24, 2006, no. 3, p. 1471
• Non-patent document 2: Yoshiyasu ITO, Katsuhiko INABA, Kazuhiko OMOTE, Yasuo WADA and Susumu IKEDA, Characterization of Submicron-scale Periodic Grooves by Grazing Incidence Ultra-small-angle X-ray Scattering, Japanese Journal of Applied Physics, Japan, The Japan Society of Applied Physics, August 10, 2007, Vol. 46, No. 32, 2007 pp. L773-L775 The U.S. 2006/0133570 A1 discloses features falling under the preamble of claim 1.

Disclosure of Invention

Meanwhile, the size of a unit structure in semiconductor manufacturing has been remarkably reduced, and it is difficult to carry out CD measurement. For example, although the beam size of a CD-SEM is considered to be around 5 nm, it is not easy to measure 20 nm with this beam size. Further, although the wavelength of measurement light is naturally reduced in scatterometry, considering the transmittance of light in the atmosphere, the wavelength is reduced by about 200 nm at most, and it is apparent that it becomes difficult to perform the measurement in the future. In the future, by reducing the CD to 32 nm, to 25 nm and then to 20 nm, i. H. over time, it is very likely that these methods will not have sufficient sensitivity. In this situation, when CD measurement using X-ray becomes possible, although not all methods such as scatterometry and CD-SEM are replaced, a new way of measuring a region having structural unit size is found at which these methods are not suitable for measuring.

Further, investigations and developments are being made on not only a semiconductor integrated circuit as a device having such a microstructure, but also a different way is being pursued which aims at high recording density of a magnetic recording medium or the like, patterned media and the like. Comparable technology using X-rays is considered applicable even for the evaluation of these devices. The above description is aimed at the background in which the present inventor started to develop the CD measurement using X-rays.

In response to the requirement for the CD measurement, by using the methods disclosed in Patent Document 2 and Non-Patent Document 1, it is possible to specify the dimension of a microstructure to some extent. However, for example, in the method disclosed in Non-patent Document 1, since the calculation is performed one-dimensionally, the heights of a structure such as a lattice are averaged in the height direction. Consequently, when the density of a sidewall portion gradually changes in line and space microstructure, it is impossible to determine whether the change is caused by the slope of the sidewall or the roughness. Like it above is described, there is a limit of accuracy in specifying the feature of a microstructure with the conventional method.

1. (1) Um die obige Aufgabe zu erzielen, wird gemäß der vorliegenden Erfindung ein Verfahren zur Messung einer Oberflächenmikrostruktur auf einer Probenoberfläche bereitgestellt, das in einem der unabhängigen Ansprüche definiert ist. Wie es oben beschrieben ist, in dem Verfahren zur Messung der Oberflächenmikrostruktur gemäß der vorliegenden Erfindung, da die Röntgenstrahlung, die eine elektromagnetische Welle ist, deren Wellenlänge ausreichend kürzer als ein zu messendes Ziel ist, zur Messung verwendet wird, ist es möglich, eine feine Struktur genauer zu messen als in einem Fall, in dem Licht oder dergleichen, deren Wellenlänge länger als das zu messende Ziel ist, verwendet wird. Es ist auch möglich, die dreidimensionalen Merkmale der periodisch angeordneten Einheitsstruktur zu ermitteln und die Oberflächenstruktur und dergleichen von verschiedenen Einrichtungen, die mit Linien und Räumen und Punkten ausgebildet sind, zu ermitteln.
2. (2) Ferner ist in dem Verfahren zur Messung einer Oberflächemikrostruktur gemäß der vorliegenden Erfindung die Einheitsstruktur mit einem gleichförmigen substantiellen Bereich und einem gleichförmigen Leerbereich innerhalb dieser Schichten ausgebildet, und die Streuintensität der Röntgenstrahlung, die von dem substantiellen Bereich bewirkt wird, wird berechnet. Wie es oben beschrieben ist, da die Merkmale, die in den Probe enthalten sind, dadurch dargestellt werden, dass diese in den substantiellen Bereich und den Leerbereich unterteilt werden, spezielle Formeln zur Berechnung der Röntgenstreuintensität bereitgestellt werden, der Fit zum Optimieren der Formparameter des substantiellen Bereichs, die dort auftreten, ausgeführt wird, ist es möglich, ein genaues Strukturmerkmale der Probe auf einfache Weise zu ermitteln.
3. (3) Ferner wird bei dem Verfahren zur Messung der Oberflächenmikrostruktur gemäß der vorliegenden Erfindung unter Berücksichtigung der Beugungs- und Reflexionseffekte, die durch eine Mehrzahl von Schichten, die in dem Probenmodell ausgebildet sind, erzeugt werden, die Streuintensität der Röntgenstreuung durch die Mikrostruktur berechnet. Folglich, da zu der Zeit das Probenmodell, in der die Schichtstruktur ausgebildet ist, angenommen bzw. festgelegt wird und die Beugungs- und Reflexionseffekte, die von diesen mehreren Schichten bewirkt werden, berücksichtigt werden, ist es möglich, eine Mikrostruktur, die auf der Oberfläche ausgebildet ist, genau zu analysieren.
4. (4) Ferner wird bei dem Verfahren zur Messung der Oberflächenmikrostruktur gemäß der vorliegenden Erfindung durch Unterstellen, dass die Einheitsstrukturen, Fluktuationen der Position von einer exakten periodischen Position aufweisen und die exakte periodische Position und die Fluktuationen der Position nicht von Differenzen zwischen wechselseitigen Positionen abhängen und zufällig sind, die Streuintensität der Röntgenstrahlung berechnet. Folglich ist es möglich, die Mikrostruktur einer Probe genau zu ermitteln, in der die Fluktuationen der Position der Einheitsstrukturen nicht von Differenzen zwischen wechselseitigen bzw. beidseitigen Positionen abhängen.
5. (5) Bei dem Verfahren zur Messung einer Oberflächemikrostruktur gemäß der vorliegenden Erfindung wird unter der Annahme, dass die Einheitsstrukturen Fluktuationen der Position bezüglich einer exakten periodischen Position aufweisen und die Fluktuationen der Position lediglich von einer relativen Positionsbeziehung bei der Einheitsstruktur abhängen, die Streuintensität der Röntgenstrahlung berechnet. Folglich ist es möglich, die Oberflächenmikrostruktur einer Probe zu ermitteln, in der die Einheitsstrukturen Fluktuationen der Position relativ zu einer exakten periodischen Position aufweisen.
6. (6) Ferner werden in dem Verfahren zur Messung einer Oberflächemikrostruktur gemäß der vorliegenden Erfindung, wenn die Fluktuationen der Position der Einheitsstrukturen eine Periodizität aufweisen, eine Amplitude und eine Periode der Fluktuationen der Position verwendet, um das mittlere Quadrat der Fluktuationen der Position der Einheitsstrukturen auszudrücken, und somit wird die Streuintensität der Röntgenstrahlung berechnet. Folglich ist es möglich, die Oberflächenmikrostruktur einer Probe zu ermitteln, in der die Fluktuationen der Position der Einheitsstrukturen eine Periodizität aufweisen.
7. (7) Ferner wird bei dem Verfahren zur Messung einer Oberflächenmikrostruktur gemäß der vorliegenden Erfindung in einem Probenmodell, in dem die Einheitsstrukturen den substantiellen Bereich in einem Zylinder aufweisen, die Streuintensität der Röntgenstrahlung berechnet. Folglich ist es möglich, die Oberflächenmikrostruktur einer Probe einfach zu messen, bei der die Einheitsstrukturen eine zylindrische Form aufweisen.
8. (8) Ferner wird bei dem Verfahren zur Messung einer Oberflächenmikrostruktur gemäß der vorliegenden Erfindung in einem Probenmodell, in dem die Einheitsstrukturen den substantiellen Bereich in einem Trapezoid aufweisen, das in einer x-Richtung parallel zur Probenoberflächen gleichförmig ist, die Streuintensität der Röntgenstrahlung berechnet. Folglich ist es möglich, die Oberflächenmikrostruktur einer Probe einfach zu messen, in der die Einheitsstrukturen, wie beispielsweise Linien und Räume, eine trapezförmige Querschnittsgestalt aufweisen, die in einer bestimmten Richtung gleichförmig ist.
9. (9) Ferner wird bei dem Verfahren zur Messung einer Oberflächenmikrostruktur gemäß der vorliegenden Erfindung bei einem Probenmodell, in dem die Einheitsstrukturen einen substantiellen Bereich aufweisen, der in einer x-Richtung parallel zur Probenoberfläche gleichförmig ist, und die in Elemente in einer y-Richtung senkrecht zur x-Richtung parallel zur Probenoberfläche unterteilt sind, ein Integral durch eine Summe der Elemente approximiert, und somit wird die Streuintensität der Röntgenstrahlung berechnet. Folglich ist es möglich, selbst detaillierte Merkmale der Oberflächenstruktur einer Probe zu ermitteln, in der die Einheitsstruktur eine Gestalt aufweist, die in einer gegebenen Richtung gleichförmig ist.
10. (10) Ferner wird bei dem Verfahren zur Messung einer Oberflächemikrostruktur gemäß der vorliegenden Erfindung, wenn ein Probenmodell, in dem die Einheitsstrukturen einen substantiellen Bereich aufweisen, der eine Querschnittsstruktur aufweist, die in der x-Richtung gleichförmig ist, angenommen wird, ist entweder ein Krümmungsradius eines konvexen Endbereichs von beiden Enden einer oberen Seite oder ein Krümmungsradius eines konkaven Basisbereichs von beiden Enden einer unteren Seite der Querschnittsgestalt in einem Parameter enthalten, und somit wird die Streuintensität der Röntgenstrahlung berechnet. Folglich ist es möglich, selbst detaillierte Merkmale, wie beispielweise den Krümmungsradius eines Endbereichs der Oberflächemikrostruktur einer Probe zu ermitteln, in der die Einheitsstruktur eine gleichförmige trapezoidförmige Querschnittsgestalt in einer gegebenen Richtung aufweist.
11. (11) Ferner wird bei dem Verfahren zur Messung einer Oberflächemikrostruktur gemäß der vorliegenden Erfindung mittels eines Probenmodells, in dem die Einheitsstrukturen einen ersten substantiellen Bereich, der in der Form eines in x-Richtung gleichförmigen Trapezoids ausgebildet ist, und einen oder mehrere zweite substantielle Bereiche aufweist, deren Material sich von der Materialt des ersten substantiellen Bereichs unterscheidet und die in Schichten auf dem ersten substantiellen Bereich ausgebildet sind, die Streuintensität der Röntgenstrahlung berechnet. Mittels Verwendung des oben beschriebenen Probenmodells ist es möglich, eine nicht destruktive Messung auszuführen, beispielsweise in dem Herstellungsprozess verschiedener Einrichtungen, wobei der zweite substantielle Bereich als ein Film auf einer Seitenwand oder einem Bodenabschnitt gleichförmig ausgebildet ist.
12. (12) Ferner wird in dem Verfahren zur Messung einer Oberflächenmikrostruktur gemäß der vorliegenden Erfindung mittels eines Probenmodells, in dem die Einheitsstrukturen den substantiellen Bereich aufweisen, der in der x-Richtung gleichförmig ist und in dem eine Querschnittsgestalt senkrecht zur x-Richtung asymmetrisch trapezoidförmig ist, die Streuintensität der Röntgenstrahlung berechnet. Bei Verwendung des oben beschriebenen Probenmodells, selbst wenn in dem Herstellungsverfahren verschiedener Einrichtungen eine asymmetrische Seitenwandstruktur ausgebildet wird, ist es möglich, eine zufriedenstellende Detektionsempfindlichkeit bezüglich der Asymmetrie eines Seitenwandwinkels zu erzielen. Folglich ist es möglich, das Probenmodell als Überwachung eines Prozesses, in dem Asymmetrie eine Rolle spielt, effektiv zu nutzen.
13. (13) Ferner wird in dem Verfahren zur Messung einer Oberflächenmikrostruktur gemäß der vorliegenden Erfindung mittels eines Probenmodells, in dem die Einheitsstrukturen einen substantiellen Bereich aufweisen, der eine periodische Struktur sowohl in einer x-Richtung parallel zur Probenoberfläche als auch einer y-Richtung parallel zur Probenoberfläche und senkrecht zur x-Richtung aufweist, und die in der x- und y-Richtung parallel zur Probenoberfläche in Elemente unterteilt sind, die Streuintensität der Röntgenstrahlung jeder der Elemente mittels einer Summe der Elemente integriert. Folglich ist es möglich, die Differenz bzw. Abweichungen einer Querschnittsgestalt zu ermitteln, selbst wenn eine zweidimensionale periodische Struktur auf der Oberfläche vorhanden ist und jede der Einheitsstrukturen eine komplizierte Querschnittsgestalt aufweist.
14. (14) Ferner wird gemäß der vorliegenden Erfindung ein Verfahren zur Datenanalyse einer Oberflächenmikrostrukturmessung zur Messung einer Mikrostruktur auf einer Probenoberfläche bereitgestellt, wobei das Verfahren einen Computer veranlasst, die Schritte auszuführen: Annehmen bzw. Festlegen eines Probenmodells mit einer Mikrostruktur auf einer Oberfläche, in der eine oder mehrere Schichten in einer Richtung senkrecht auf der Oberfläche ausgebildet sind und Einheitsstrukturen periodisch in einer Richtung parallel zur Oberfläche innerhalb der Schichten angeordnet sind, Berechnen einer Streuintensität von Röntgenstrahlung, die von der Mikrostruktur gestreut wird, unter Berücksichtigung von Beugungs- und Reflexionseffekten, die von den Schichten erzeugt werden, und Durchführen eines Fits der aus dem Probenmodell berechneten Streuintensität der Röntgenstrahlung an eine Streuintensität, die durch Bestrahlen der Probenoberfläche mit Röntgenstrahlung in einem sehr kleinen bzw. streifenden Einfallswinkel tatsächlich gemessen wird; und Bestimmen, als Resultat des Fits, eines optimalen Parameterwerts zur Spezifizierung einer Gestalt bzw. Form der Einheitsstrukturen.

The present invention has been made in view of the above situation, and an object of the present invention is to provide a surface microstructure measurement method, a surface microstructure measurement data analysis method and an X-ray diffraction measuring device, which can accurately measure a microstructure on a surface can, and with which a three-dimensional structural feature can be determined.

1. (1) In order to achieve the above object, according to the present invention, there is provided a method for measuring a surface microstructure on a sample surface as defined in any one of the independent claims. As described above, in the method for measuring the surface microstructure according to the present invention, since the X-ray, which is an electromagnetic wave whose wavelength is sufficiently shorter than a target to be measured, is used for measurement, it is possible to obtain a fine to measure structure more accurately than in a case where light or the like whose wavelength is longer than the target to be measured is used. It is also possible to detect the three-dimensional characteristics of the unit structure periodically arranged and to detect the surface structure and the like of various devices formed with lines and spaces and dots.
2. (2) Further, in the method for measuring a surface microstructure according to the present invention, the unit structure having a uniform substantial area and a uniform void area is formed within these layers, and the X-ray scattering intensity caused by the substantial area is calculated. As described above, since the features contained in the specimen are represented by dividing them into the substantial area and the void area, specific formulas for calculating the X-ray scattering intensity are provided, the fit for optimizing the shape parameters of the substantial Areas that occur there, it is possible to determine an accurate structural features of the sample in a simple manner.
3. (3) Further, in the surface microstructure measuring method according to the present invention, considering the effects of diffraction and reflection produced by a plurality of layers formed in the sample model, the scattering intensity of X-ray scattering by the microstructure is calculated. Consequently, since at the time the sample model in which the layered structure is formed is assumed and the diffraction and reflection effects caused by these multiple layers are taken into account, it is possible to obtain a microstructure formed on the surface trained to analyze accurately.
4. (4) Further, in the method for measuring the surface microstructure according to the present invention, by assuming that the unit structures have fluctuations in position from an exact periodic position and the exact periodic position and the fluctuations in position do not depend on differences between mutual positions and are random, the scattering intensity of the X-ray radiation is calculated. Consequently, it is possible to accurately detect the microstructure of a sample in which the fluctuations in the position of the unit structures do not depend on differences between mutual positions.
5. (5) In the method for measuring a surface microstructure according to the present invention, assuming that the unit structures have fluctuations in position with respect to an exact periodic position and the fluctuations in position depend only on a relative positional relationship in the unit structure, the scattering intensity of X-rays calculated. Consequently, it is possible to detect the surface microstructure of a sample in which the unit structures have fluctuations in position relative to an accurate periodic position.
6. (6) Further, in the method for measuring a surface microstructure according to the present invention, when the fluctuations in the position of the unit structures have a periodicity, an amplitude and a period of the fluctuations in the position are used to express the mean square of the fluctuations in the position of the unit structures , and thus the scattering intensity of the X-ray radiation is calculated. Consequently, it is possible to detect the surface microstructure of a sample in which the fluctuations in the position of the unit structures have periodicity.
7. (7) Further, in the method for measuring a surface microstructure according to the present invention, in a sample model in which the unit structures have the substantial region in a cylinder, the X-ray scattering intensity is calculated. Consequently, it is possible that Easy to measure surface microstructure of a sample in which the unit structures have a cylindrical shape.
8. (8) Further, in the method for measuring a surface microstructure according to the present invention, in a sample model in which the unit structures have the substantial area in a trapezoid that is uniform in an x-direction parallel to the sample surface, the X-ray scattering intensity is calculated. Consequently, it is possible to easily measure the surface microstructure of a sample in which the unit structures such as lines and spaces have a trapezoidal cross-sectional shape that is uniform in a certain direction.
9. (9) Further, in the method for measuring a surface microstructure according to the present invention, in a sample model in which the unit structures have a substantial area uniform in an x-direction parallel to the sample surface and the elements in a y-direction perpendicular to the x-direction parallel to the sample surface, an integral is approximated by a sum of the elements, and thus the scattering intensity of the X-ray is calculated. Consequently, it is possible to detect even detailed features of the surface structure of a sample in which the unit structure has a shape uniform in a given direction.
10. (10) Further, in the method for measuring a surface microstructure according to the present invention, when a sample model in which the unit structures have a substantial area having a cross-sectional structure that is uniform in the x-direction is assumed to be either a Radius of curvature of a convex end portion from both ends of an upper side or a radius of curvature of a concave base portion from both ends of a lower side of the cross-sectional shape is included in a parameter, and thus the scattering intensity of the X-ray is calculated. Consequently, it is possible to detect even detailed features such as the radius of curvature of an end portion of the surface microstructure of a sample in which the unit structure has a uniform trapezoidal cross-sectional shape in a given direction.
11. (11) Further, in the method for measuring a surface microstructure according to the present invention, using a sample model in which the unit structures include a first substantial region formed in the shape of a trapezoid uniform in the x-direction and one or more second substantial regions the material of which is different from the material of the first substantial region and which are formed in layers on the first substantial region, calculates the scattering intensity of the X-ray. By using the sample model described above, it is possible to perform non-destructive measurement, for example, in the manufacturing process of various devices, with the second substantial region being uniformly formed as a film on a side wall or a bottom portion.
12. (12) Further, in the method for measuring a surface microstructure according to the present invention, using a sample model in which the unit structures have the substantial area that is uniform in the x-direction and in which a cross-sectional shape perpendicular to the x-direction is asymmetrically trapezoidal , calculates the scattering intensity of X-rays. Using the sample model described above, even if an asymmetric sidewall structure is formed in the manufacturing process of various devices, it is possible to obtain a satisfactory detection sensitivity to the asymmetry of a sidewall angle. Consequently, it is possible to effectively use the sample model as a monitor of a process in which asymmetry plays a role.
13. (13) Further, in the method for measuring a surface microstructure according to the present invention, using a sample model in which the unit structures have a substantial portion having a periodic structure in both an x-direction parallel to the sample surface and a y-direction parallel to the sample surface and perpendicular to the x-direction, and which are divided into elements in the x- and y-directions parallel to the sample surface, the scattering intensity of the X-ray of each of the elements is integrated by means of a sum of the elements. Consequently, it is possible to detect the difference of a cross-sectional shape even when a two-dimensional periodic structure is present on the surface and each of the unit structures has a complicated cross-sectional shape.
14. (14) Further according to the present invention, there is provided a surface microstructure measurement data analysis method for measuring a microstructure on a sample surface, the method causing a computer to perform the steps of: accepting or specifying a sample model having a microstructure on a surface in which one or more layers are formed in a direction perpendicular to the surface and unit structures are periodically formed in a direction parallel to the surface within the layers, calculating a scattering intensity of X-rays scattered by the microstructure, taking into account diffraction and reflection effects generated by the layers, and performing a fit of the X-ray scattering intensity calculated from the sample model a scattering intensity actually measured by irradiating the sample surface with X-rays at a very small or grazing angle of incidence; and determining, as a result of the fitting, an optimal parameter value for specifying a shape of the unit structures.

• (15) Ferner wird gemäß der vorliegenden Erfindung eine Röntgenstreuungs-Messeinrichtung bereitgestellt, die zur Messung einer Mikrostruktur auf einer Probenoberfläche ausgelegt ist, wobei die Röntgenstreuungs-Messeinrichtung enthält: einen Monochromator, der Röntgenstrahlung, die von einer Röntgenstrahlquelle emittiert wird, spektral reflektiert; einen Schlitzabschnitt, der im Stande ist, eine Spot-Abmessung der spektral reflektierten Röntgenstrahlung auf der Probenoberfläche auf 30 µm oder weniger zu begrenzen; einen Probenhalter, der eine Drehung zum Ändern sowohl eines Einfallwinkels der spektral reflektierten Röntgenstrahlung auf der Probenoberfläche als auch eine Drehung in einer Ebene der Probenoberfläche ermöglicht und der die Probe trägt; und einen zweidimensionalen Detektor, der die Streuintensität der Röntgenstrahlung, die von der Probenoberfläche gestreut wird, misst. Da es auf diese Weise möglich ist, die Ausdehnung des Bestrahlungsbereichs der Röntgenstrahlung auf die Probenoberfläche einzuschränken und die Streuintensität der Röntgenstrahlung zu messen, welche eine Mikrostruktur in der Größenordnung von Nanometern, die auf der Oberfläche ausgebildet ist, widerspiegelt, ist eine genaue Messung der Mikrostruktur möglich.

Consequently, an advantage is obtained by using short-wavelength X-rays, and it is possible to accurately measure a microstructure compared with a case where light scattering intensity or the like is used. Further, using the sample model in which a layered structure is formed and the scattering intensity corresponding to the shape of the substantial area on the surface, it is possible to detect a three-dimensional structural feature of the sample and the surface structure and the like of various devices obtained by Lines and spaces and points formed will determine.

• (15) Further according to the present invention, there is provided an X-ray scattering measuring device designed to measure a microstructure on a sample surface, the X-ray scattering measuring device including: a monochromator that spectrally reflects X-rays emitted from an X-ray source; a slit portion capable of restricting a spot size of the spectrally reflected X-ray on the sample surface to 30 µm or less; a sample holder that allows rotation for changing both an incident angle of the spectrally reflected X-ray on the sample surface and rotation in a plane of the sample surface and supports the sample; and a two-dimensional detector that measures the scattering intensity of X-rays scattered from the sample surface. In this way, since it is possible to restrict the extension of the irradiation range of X-rays to the sample surface and to measure the scattering intensity of X-rays reflecting a microstructure on the order of nanometers formed on the surface, accurate measurement of the microstructure is possible possible.

According to the present invention, it is possible to accurately measure a microstructure compared to a case where light scattering or the like is used. With the sample model in which a layered structure is formed and the scattering intensity corresponding to the shape of the substantial area on the surface, it is possible to determine the three-dimensional structural feature of the sample, and the surface structure and the like of various facilities represented by lines and spaces and points is designed to determine.

character list

• 1 Fig. 14 is a diagram showing the structure and functional blocks of a surface microstructure measuring system according to the present invention;
• 2 Fig. 14 is a side view showing part of an example of construction of an X-ray diffraction measuring device according to the present invention;
• 3 Fig. 14 is a side view showing part of an example of the construction of the X-ray diffraction measuring device according to the present invention;
• 4 Fig. 13 is a perspective view (or an additional diagram illustrating a formula for the simulation) schematically showing an example of a sample having a surface microstructure;
• 5 Fig. 13 is a perspective view (or an additional diagram illustrating a formula for the simulation) schematically showing an example of a sample having a surface microstructure;
• 6 Fig. 12 is a flow chart showing the flow of a simulation and a fit in a surface microstructure measuring device;
• 7 Fig. 12 is a schematic diagram showing how an electric field changes by the incidence of X-rays in layers of a sample having an N-layer structure;
• 8th Fig. 12 is a schematic diagram showing how the electric field changes by the exiting X-rays in the layers of the sample having the N-layer structure;
• 9 Fig. 14 is a cross-sectional view of a sample model in which lines and spaces are formed on the surface thereof;
• 10 Fig. 14 is a cross-sectional view showing a sample model in which a layered structure is formed;
• 11 Fig. 14 is a cross-sectional view showing a sample model having a structure in which the material composition changes as the height changes;
• 12 Fig. 14 is a cross-sectional view showing a sample model in which a convex portion has a step;
• 13 Fig. 14 is a cross-sectional view showing a sample model in which a new resist layer is formed on a microstructure;
• 14 Fig. 12 is a diagram showing a scattering intensity derived from a part of the sum of the periods calculated with a spurious period of p=2;
• 15 Fig. 14 is a cross-sectional view of a sample model having a line portion whose cross-sectional view is trapezoidal;
• 16 Fig. 12 is a diagram showing the result of a simulation performed on the model of 15 was executed;
• 17 Fig. 14 is a cross-sectional view showing a sample model having a line portion whose cross-sectional view is trapezoidal;
• 18 Fig. 12 is a diagram showing the result of a simulation performed on the model of 17 was executed;
• 19 Fig. 14 is a cross-sectional view of a sample model having a line portion whose surface is covered with layers;
• 20 Fig. 12 is a diagram showing the result of a simulation performed on the model of 19 was executed;
• 21 Fig. 14 is a cross-sectional view showing a sample model having a line portion in which an asymmetric side wall is formed;
• 22 Fig. 12 is a diagram showing the result of a simulation performed on the model of 21 was executed;
• 23 Fig. 12 is a top view SEM photograph of a sample;
• 24 Fig. 12 is a cross-sectional TEM photograph of a sample;
• 25 Fig. 12 is a cross-sectional view of the sample model of the samples shown in Figs 23 and 24 are shown;
• 26 Fig. 14 is a graph showing an X-ray scattering intensity actually measured;
• 27 Fig. 14 is a graph showing a calculated X-ray scattering intensity;
• 28 Fig. 14 is a graph showing actually measured X-ray scattering intensity and calculated X-ray scattering intensity with respect to the first peak;
• 29 Fig. 14 is a graph showing actually measured X-ray scattering intensity and calculated X-ray scattering intensity with respect to the third peak; and
• 30 12 is a plan view showing an example of a semiconductor substrate.

Best Modes for Carrying Out the Invention

An embodiment of the present invention will be described below with reference to the accompanying drawings. For ease of understanding of the description, the same components in the drawings are denoted by common reference symbols and their description is not repeated.

[Structure of the overall system]

1 FIG. 12 is a diagram showing the structure and functional blocks of a surface microstructure measurement system 100. FIG. like it in 1 As shown, the surface microstructure measurement system 100 is equipped with an X-ray scattering measurement device 110 and a surface microstructure analysis device 120 .

The X-ray scattering measuring device 110 is a device which irradiates a sample with X-rays at a small angle and which can measure the scattering intensity. The surface microstructure analyzer 120 is a device that uses a known parameter to calculate the scattering intensity and can calculate a feature of the surface microstructure of the sample by performing a fit to an actually measured value. The surface microstructure analyzer 120 is a computer including a CPU, a storage device, an input device, and an output device like a PC or the like, and the input device and the output device may be provided externally. The input device is, for example, a keyboard or a mouse and is used when a known parameter or the like is input. The output device is a display or a printer, for example, and outputs the result of the fit.

The surface microstructure analyzer 120 is connected to the X-ray scattering measuring device 110 and stores measurement data automatically transmitted from the X-ray scattering measuring device 110 . Preferably, the data is automatically transmitted, but the data may be stored in the surface microstructure analyzer 120 on a recording medium or the like. Alternatively, a control program is installed in advance, and thus it is possible to control the X-ray scattering measurement device 110 via the surface microstructure analysis device 120 at the time of actual measurement.

[Structure of Analyzer]

The surface microstructure analyzer 120 includes a parameter acquisition section 121, a formula storage section 122, a simulation section 123, a fitting section 124 and an output section 125. The parameter acquisition section 121 acquires parameters for specifying conditions of X-ray scattering, which are obtained from the X-ray scattering measuring apparatus 110 and parameters entered by a user. The acquired parameters include, for example, an angle of incidence α on the sample 140 of the X-ray and an initial value of a parameter for specifying the shape of a unit structure on the surface of the sample. Parameters obtained by the fit include, for example, the height H of the sample whose cross-sectional view is trapezoidal, the length Wt of the upper side, the length Wb of the lower side, the radius of curvature Rt of a convex end portion of the upper side, the radius of curvature Rb of a convex broad base portion (edge of lower portion).

The formula storage section 122 stores a formula for calculating a scattering intensity on a specific sample model through simulation. On the other hand, the simulation section 123 obtains from the formula storage section 122 a formula for calculating the scattering on the desired sample model from the formula storage section 122, and on the other hand, the simulation section 123 selects various necessary parameter values from known parameters obtained from the known parameters and calculates an X-ray scattering intensity. The fitting section 124 fits the X-ray scattering intensity calculated by the simulation section 123 and the X-ray scattering intensity actually measured by the X-ray scattering measuring device 110 .

When each formula is used to calculate the X-ray scattering intensity, various numbers such as an angle of incidence α, the refractive index of the m-th layer, and a polarization factor P are required. For example, the angle of incidence α is obtained by automatic transmission through the X-ray scattering measuring device 110, the refractive index n _m of the mth layer and the polarization factor P are obtained by manual input, and the classical electron radius r _c is obtained by applying information stored in advance is obtained. To do this, the surface microstructure analyzer 120 needs an input unit, a storage unit and the like, and based on values obtained from these various units, the simulation section 123 calculates the scattering intensity. The calculation of the X-ray scattering intensity and the operation of the surface microstructure analyzer 120 at the time of fitting will be described later. the Bre The refractive index _nm , which agrees with the structural model, can be determined from the measured X-ray scattering intensity.

[Structure of the measuring device]

2 and 3 12 are side views showing part of an example of the structure of the X-ray diffraction measuring device 110. FIG. In the construction of 2 For example, the X-ray scattering measuring device 110 includes a monochromator 113, a first collimation block 114, a second collimation block 115, a sample holder 115a, a two-dimensional detector 116 and a beam stopper 117.

The monochromator 113 spectrally reflects the X-ray emitted from an unillustrated X-ray source and illuminates the sample 140 with the spectrally reflected X-ray. The first collimation block 114 and the second collimation block 115 are formed with a member capable of interrupting the X-ray and form a slit portion that narrows the spectrally reflected X-ray. With this structure, an angle at which the sample 140 is irradiated with the X-ray is reduced to a range of 0.1° or more and 0.5° or less. With the pair of collimation blocks 114 and 115, it is possible to limit the range of X-rays on the sample surface to 30 µm or less. In this way, since it is possible to limit the extension of the irradiation range of X-rays to the sample surface and to measure the scattering intensity of X-rays reflecting a nanometer-order microstructure formed on the surface is a precise measurement of the microstructure possible. It should be noted that the structure which limits the range to 20 µm or less is more preferable. As described above, by reducing the spot and the incident angle, it is possible to measure a nanometer-order microstructure on the sample surface. In particular, by using the collimation blocks 114 and 115, it is possible to precisely interrupt the X-ray and increase the precision of the collimation.

The sample holder 115 supports the sample 140 on a flat mount. The sample holder 115 can rotate to change the angle of incidence of the spectrally reflected X-rays on the sample surface and can rotate in the plane of the sample surface. As described above, the sample 140 can rotate, and hence it is possible to measure the intensity of scattering by the sample 140 according to the diffraction angle.

The sample 140 is an element that has a microstructure on the surface; for example, the sample 140 is a substrate formed of silicon or the like and having a surface microstructure. On the detection surface, the two-dimensional detector 116 measures the scattered intensity of the X-ray radiation that is scattered on the sample surface. The beam stopper 117 receives the incident X-rays that have passed through the sample 140 . As described above, the X-ray diffraction measuring device 110 has a structure suitable for measuring the microstructure of the sample surface.

In the structure that 3 1, the X-ray scattering measuring device 110 includes a slit 118 and a sharp edge 119 instead of the pair of collimating blocks 114 and 115 in the structure shown in FIG 2 is shown. As described above, with the slit 118 and the sharp edge 119, it is possible to easily adjust the X-ray spot size.

[Rehearse]

The 4 and 5 12 are perspective views schematically showing examples of the sample 140 and a sample 145 having a surface microstructure. Samples 140 and 145 have minute periodic structural units from a few nanometers to a few hundred nanometers on their surfaces. These samples are irradiated with X-rays, whereby the scattered X-rays are measured and analyzed, and thus it is possible to measure parameters characterizing the periodically provided unit structures.

The sample 140, which in 4 shown is in the form of a substrate; a microstructure in which, on the surface 141, cylindrical unit structures are periodically arranged in the x-direction and the y-direction of the figure is formed. Likewise, sample 145, found in 5 is shown formed in the form of a substrate; on the surface 146 are unit structures, which are in of the x-direction and whose cross section is rectangular with respect to a yz-plane are periodically arranged in the y-direction.

For these samples 140 and 145, the X-rays whose spot size is narrowed to 50 μm or less, and preferably 30 μm or less, impinge on the surface 141 at an angle of incidence α, being a fit with respect to the intensity of the actually measured scattered X-rays and the intensity of the scattered X-ray calculated using a sample model of the same shape, and thus it is possible to obtain an actual sample size. When an X-ray experiment is actually carried out, the samples 140 and 145 are arranged so that the direction of the periodic structure coincides with the direction of the incident X-ray. In the figures, the x-direction is a direction in which the sample surface when the sample is placed and the incident surface of the X-ray intersect, the z-direction being a direction perpendicular to the sample surface and the y-direction is a direction perpendicular to both the x-direction and the z-direction.

[measurement method]

The method of measuring the surface microstructure with the surface microstructure measuring system 100 constructed as described above will be described below. The designated sample is first set on the sample holder to coincide with the direction of the microstructure of the sample, the sample surface is irradiated with X-rays at a grazing incidence angle α, and thus the X-ray scattering intensity is measured. The X-ray scattering intensity is measured according to an X-ray exit angle β. In order to utilize the diffraction of X-rays due to the periodic structure, the sample is measured while rotating the sample about the z-axis in the plane, if necessary.

Then, the sample model is determined using parameters specifying the shape of the unit structure having the periodic structure of the specific sample, and the X-ray scattering intensity is calculated by simulation. Specifically, one or more layers are formed in the direction perpendicular to the surface in the microstructure on the surface, and the sample model in which the unit structure is periodically arranged in the direction parallel to the surface in the layers is determined , wherein the scattering of the X-rays, which is diffracted and reflected by their interfaces, caused by the structures, is calculated and based on this a fit is performed between the X-ray scattering intensity calculated from the sample model and the measured scattering intensity. As a result of the fitting, the optimal values of the parameters specifying the shape of the unit structure are determined. A detailed description is given below.

[Formulas for the simulation]

〈FDWSA〉 :

Mittelwert von F^DWBA bezüglich Schwankungen bzw. Abweichungen zwischen den Struktureinheiten j

〈|FDWBA|2〉 :

Mittelwert von |F^DWBA|² bezüglich Schwankungen bzw. Abweichungen zwischen den Struktureinheiten j

$$F_{mj}\left(Q_z^a,Q_{//}\right)\equiv {\displaystyle \underset{s}{\int }\frac{e^{-i{Q}_z^a{Z}_{mj}\left(x_j,y_j\right)}-1}{-i{Q}_z^a}}{e}^{-i\left(Q_x\cdot x_j+Q_y\cdot y_j\right)}d{x}_{j}d{y}_j$$

$$Q_{//}=\left(Q_x,Q_y\right)$$

$$Q_{zm}{}^a=\left\{Q_{zm}{}^{T\tilde{T}},Q_{zm}{}^{R\tilde{T}},Q_{zm}{}^{T\tilde{R}},Q_{zm}{}^{R\tilde{R}}\right\}$$

$$Q_{zm}{}^{T\tilde{T}}=k_0\left({\eta }_m+{\zeta }_m\right)$$

$$Q_{zm}{}^{R\tilde{T}}=k_0\left(-{\eta }_m+{\zeta }_m\right)$$

$$Q_{zm}{}^{T\tilde{R}}=k_0\left({\eta }_m-{\zeta }_m\right)$$

$$Q_{zm}{}^{R\tilde{R}}=k_0\left(-{\eta }_m-{\zeta }_m\right)$$

$${\eta }_m=\sqrt{n_m-{\text{<figure-callout id="2" label="cos" filenames="DE112010001894B4\_0093.png,DE112010001894B4\_0094.png" state="{{state}}">cos</figure-callout>}}^2\alpha }$$

$${\zeta }_m=\sqrt{n_m-{\text{<figure-callout id="2" label="cos" filenames="DE112010001894B4\_0093.png,DE112010001894B4\_0094.png" state="{{state}}">cos</figure-callout>}}^2\beta }$$

α

Einfallswinkel

β

Ausgangswinkel

nm

Brechungsindex der m-ten Schicht

Q//

Komponentenvektor parallel zur Oberfläche des Streuvektors

rc

klassischer Elektronenradius

P

Polarisationsfaktor

N

Gesamtzahl der Einheitsstrukturen

Fj

Formfaktor der Einheitsstruktur

X̅j

ideale periodische Position der Einheitsstrukturen

u(Xj)

Versetzung der Position der Einheitsstruktur j aufgrund von lokaler Störung

x

Richtung, in der die Einfallsebene und die Probenoberfläche einander schneiden

y

Richtung senkrecht auf der x-Richtung und der z-Richtung

z

Richtung senkrecht auf der Probenoberfläche

FDWBA

basiert auf der Gleichung (37), die später beschrieben wird.

The equations for the simulation for each sample model are described. For the sample model that has a layered structure on the surface and that has a microstructure in which the unit structures in the layers are arranged periodically, the X-ray scattering intensity can be calculated using Equation (1) below. Here they will 4 and 5 used as additional diagrams to aid in understanding the equations below.
[Formula 1] $$I\left(a,\beta ,Q_{//}\right)={\left({\mathrm{right}}_{c}P\right)}^{2}N\left\{〈{\left|f^{DWBA}\right|}^2〉+{\left|〈f^{DWBA}〉\right|}^2{\displaystyle \sum _{\left|k\right|\ge 1}{〈e^{-i{Q}_{//}\left(\mathrm{and}\left(X_j\right)-\mathrm{and}\left(X_k\right)\right)}〉}_j{e}^{-i{Q}_l\cdot \left({\overline{X}}_j-{\overline{X}}_{j+k}\right)}}\right\}$$

〈FDWSA〉 :

Mean value of F ^DWBA with regard to fluctuations or deviations between the structural units j

〈|FDWBA|2〉 :

Mean of |F ^DWBA | ² with regard to fluctuations or deviations between the structural units j

$$f_{mj}\left(Q_{\mathrm{e.g}}^a,Q_{//}\right)\equiv {\displaystyle \underset{s}{\int }\frac{e^{-i{Q}_{\mathrm{e.g}}^a{Z}_{mj}\left(x_j,y_j\right)}-1}{-i{Q}_{\mathrm{e.g}}^a}}{e}^{-i\left(Q_x\cdot x_j+Q_y\cdot y_j\right)}\mathrm{i.e}{x}_j\mathrm{i.e}{y}_j$$

$$Q_{//}=\left(Q_x,Q_y\right)$$

$$Q_{\mathrm{e.g}m}{}^a=\left\{Q_{\mathrm{e.g}m}{}^{T\tilde{T}},Q_{\mathrm{e.g}m}{}^{R\tilde{T}},Q_{\mathrm{e.g}m}{}^{T\tilde{R}},Q_{\mathrm{e.g}m}{}^{R\tilde{R}}\right\}$$

$$Q_{\mathrm{e.g}m}{}^{T\tilde{T}}=k_0\left(n_m+{\zeta }_m\right)$$

$$Q_{\mathrm{e.g}m}{}^{R\tilde{T}}=k_0\left(-n_m+{\zeta }_m\right)$$

$$Q_{\mathrm{e.g}m}{}^{T\tilde{R}}=k_0\left(n_m-{\zeta }_m\right)$$

$$Q_{\mathrm{e.g}m}{}^{R\tilde{R}}=k_0\left(-n_m-{\zeta }_m\right)$$

$$n_m=\sqrt{n_m-{\text{cos}}^{2}a}$$

$${\zeta }_m=\sqrt{n_m-{\text{<figure-callout id="2" label="cos" filenames="DE112010001894B4\_0093.png,DE112010001894B4\_0094.png" state="{{state}}">cos</figure-callout>}}^2\beta }$$

a

angle of incidence

β

exit angle

nm

Refractive index of the mth layer

Q//

Component vector parallel to the surface of the scattering vector

RC

classical electron radius

P

polarization factor

N

Total number of unit structures

fj

Unit structure form factor

X̅j

ideal periodic position of the unit structures

u(Xj)

Shift in position of unit structure j due to local disturbance

x

Direction in which the plane of incidence and the sample surface intersect

y

Direction perpendicular to the x-direction and the z-direction

e.g

Direction perpendicular to the sample surface

FDWBA

is based on equation (37) which will be described later.

In the equation (1), it is assumed that the unit structure is formed by a substantial area and a void area uniform in the layers and that the substantial area causes the scattering, and Zmj (xj, yj) denotes the boundary between the substantial area and the blank space. As described above, using the scattering intensity equation that corresponds to the sample model in which the layered structure is formed and the shape of the substantial area on the surface, it is possible to determine a three-dimensional structural feature of the sample and the surface structure and to determine the like of various entities formed by lines and spaces and points.

Qu

Projektion des Streuvektors in der u-Richtung

Δu̅

Mittelwert der Fluktuationen der Positionen

In the equation (1), although the fluctuations of the position u(X _j ) of the unit structure from the exact periodic position are taken into account, in the form it is impossible to perform a specific calculation on u(X _j ). If it can be assumed that the fluctuations in the position of the unit structure do not depend on Xj and are random, Equation (2) below can be used.
[Formula 2] $$I\left(a,\beta ,Q_{//}\right)={\left({\mathrm{right}}_{c}P\right)}^{2}N\left\{〈{\left|f^{DWBA}\right|}^2〉+{\left|〈f^{DWBA}〉\right|}^2{e}^{\frac{-Q_{\mathrm{and}}^2\Delta {\overline{\mathrm{and}}}^2}{2}}{\displaystyle \sum _{\left|k\right|\ge 1}{e}^{-i{Q}_{//}\cdot \left({\overline{X}}_j-{\overline{X}}_{j+k}\right)}}\right\}$$

Qu

Projection of the scattering vector in the u-direction

Δu̅

Average of the fluctuations of the positions

In the equation (2), it is assumed that the effects of the fluctuations in the positions on the scattering do not depend on the relative positional relationship between the unit structures. Consequently, it is possible to detect the surface microstructure of the sample in which the unit structures have random fluctuations in positions.

Qu

Projektion des Streuvektors in der u-Richtung

b

Amplitude der Fluktuationen der Positionen

p

Periode der Fluktuationen der Positionen

ΔXk

Abstand zwischen den Einheitsstrukturen

On the other hand, if the sample model having a periodicity of the fluctuations of the positions of the unit structure can be assumed, Equation (3) below can be used.
[Formula 3] $$I\left(a,\beta ,Q_{//}\right)={\left({\mathrm{right}}_{c}P\right)}^{2}N\left\{〈{\left|f^{DWBA}\right|}^2〉+{\left|〈f^{DWBA}〉\right|}^2{\displaystyle \sum _{\left|k\right|\ge 1}{e}^{-i{Q}_{//}\left({\overline{X}}_j-{\overline{X}}_{j+k}\right)}{e}^{-\frac{Q_{\mathrm{and}}^2}{2}{a}^2{\left(1-\text{cos}\left(2\pi \frac{\left|\mathrm{\Delta }{X}_k\right|}{P}\right)\right)}^2}}\right\}$$

Qu

Projection of the scattering vector in the u-direction

b

Amplitude of position fluctuations

p

Period of fluctuations in positions

ΔXk

Distance between the unit structures

As described, when the fluctuations in the positions of the unit structure have a periodicity, it is possible to calculate the X-ray scattering intensity by using the amplitude and the period of the fluctuations in the positions and to obtain the surface microstructure of the sample.

Although in order to integrate the form factor F _j of the unit structure, it is necessary to determine an integral by introducing a parameter according to the shape of the unit structure of the sample model in a specific case, it is possible to determine an integral using a simple equation. The form factor used with the shape of the unit structure is described below.

(cylindrical shape)

A

Radius der zylindrischen Form der Einheitsstruktur

H

Höhe der zylindrischen Form der Einheitsstruktur

J1

Vesselfunktion

If a sample model in which the unit structure has a cylindrical shape is assumed, the form factor F _j of the unit structure shown in Equation (4) below can be used. In the equation (4), a sample model in which the unit structure has a cylindrical substantial portion is assumed.
[Formula 4] $$f_j\left(Q_{\mathrm{e.g}}^a,{\text{Q}}_{//}\right)=2\pi A\frac{e^{-i{Q}_{\mathrm{e.g}}^{a}H}-1}{-i{Q}_{\mathrm{e.g}}^a}\frac{J_1\left(A\cdot \left|{\text{Q}}_{//}\right|\right)}{\left|{\text{Q}}_{//}\right|}$$

A

Radius of the cylindrical shape of the unit structure

H

Height of the cylindrical shape of the unit structure

J1

vessel function

(shape with trapezoidal cross section)

d(x)

d-Funktion

Qx

x-Richtungskomponente eines Streuvektors

Qy

y-Richtungskomponente des Streuvektors

Wt

Länge einer oberen Seite eines Trapezoidquerschnitts der Einheitsstruktur

Wb

Länge einer unteren Seite des trapezoidförmigen Querschnitts der Einheitsstruktur

H

Länge des trapezoidförmigen Querschnitts der Einheitsstruktur

If a sample model in which the unit structure has a shape whose cross section in the x-direction is uniformly trapezoidal can be assumed, the shape factor F _j of the unit structure represented by Equation (5) below can be used. In Equation (5), a sample model is assumed in which the unit structure has a trapezoidal substantial area uniform in the x-direction parallel to the sample surface.
[Formula 5] $$f_j\left(Q_{\mathrm{e.g}}^a,Q_{//}\right)=2\pi \delta \left(Q_x\right)\frac{1}{i{Q}_y}\left[e^{-i\frac{W_b}{2}}{Q}_y\frac{e^{-i\left(Q_{\mathrm{e.g}}^a-\frac{W_b-W_t}{2H}{Q}_y\right)H}-1}{-i\left(Q_{\mathrm{e.g}}^a-\frac{W_b-W_t}{2H}{Q}_y\right)}-e^{i\frac{Wb}{2}{Q}_y}\frac{e^{-i\left(Q_{\mathrm{e.g}}^a+\frac{W_b-W_t}{2H}{Q}_y\right)H}-1}{-i\left(Q_{\mathrm{e.g}}^a+\frac{W_b-W_t}{2H}{Q}_y\right)}\right]$$

d(x)

d function

Qx

x-direction component of a scattering vector

qy

y-direction component of the scattering vector

Wt

Length of an upper side of a trapezoidal cross section of the unit structure

wb

Length of a lower side of the trapezoidal cross section of the unit structure

H

Length of the trapezoidal cross-section of the unit structure

(Other Complicated Shapes)

In a case where a shape uniform in the x-direction is used as a target, if the simple shape sample model described above is used, it is possible to mathematically determine F _j .

$$\mathrm{\Delta }y=\frac{L}{n}$$

$$\mathrm{\Delta }Qy=\frac{2\pi }{L}$$

d(x)

d-Funktion

Qx

x-Richtungskomponente eines Streuvektors

Qy

y-Richtungskomponente des Streuvektors

L

Länge der Einheitsstruktur in der y-Richtung

n

Anzahl der unterteilten Elemente der Einheitsstruktur in der Y-Richtung

h

Streuordnung, wenn L ein Oberflächenraum ist

However, when a sample model of a complicated shape has to be adopted, such as when the radius of curvature of an end portion has to be taken into account in a shape whose cross section is trapezoidal, a sample model in which the unit structure has a uniform shape in the x-direction and a division of the elements in the y-direction is performed, and it is thus possible to perform a fit using the form factor F _j of the unit structure represented by equation (6) below. With such an equation, it is possible to obtain parameters such as the height of a trapezoidal cross section, an upper side, a lower side, the radius of curvature of both ends of the upper side, and the radius of curvature of the base portion of both ends of the lower side. As described above, Equation (6) relates to a sample model in which the unit structure has a uniform substantial area in the x-direction parallel to the sample surface and in which the subdivision of the elements in the y-direction parallel to the sample surface and is made perpendicular to the x-direction. The sum of the elements is used to approximate the integral.
[Formula 6] $$f_j\left(Q_{\mathrm{e.g}}^a,Q_x,Q_y\right)=2\pi \delta \left(Q_x\right)\cdot \mathrm{\Delta }y{\displaystyle \sum _{s=\frac{n-1}{2}}^{s=\frac{n-1}{2}}\frac{e^{-i{Q}_{\mathrm{e.g}}^{a}Z\left(\mathrm{\Delta }y\cdot s\right)}-1}{-i{Q}_{\mathrm{e.g}}^a}{e}^{-2\pi i\frac{J_{t\cdot s}}{n}}}$$

$$\mathrm{\Delta }y=\frac{L}{n}$$

$$\mathrm{\Delta }Qy=\frac{2\pi }{L}$$

d(x)

d function

Qx

x-direction component of a scattering vector

qy

y-direction component of the scattering vector

L

Length of the unit structure in the y-direction

n

Number of divided elements of the unit structure in the Y-direction

H

Scattering order when L is a surface space

In the equation (6), with respect to the yz cross section of the unit structure, the height, the length of an upper side, the length of a lower side, the radius of curvature of a convex end portion of both ends of the upper side and the radius of curvature of the concave base portion of both ends of the lower side can be used as parameters identifying the unit structure. As described above, with Equation (6), it is possible to determine even detailed features.

[Calculation method (simulation and fit)]

Next, the simulation and the fitting will be described as methods for determining the X-ray scattering intensity by using parameters specifying the unit structures formed in the sample layers having a periodic structure. 6 FIG. 12 is a flowchart showing the flow of simulation and fitting in the surface microstructure analyzer 120. FIG. An X-ray scattering intensity measured in advance and actually measured is set to be automatically transmitted from the X-ray scattering device 110 and stored in the surface microstructure analyzer 120 .

First, a sample model is adopted according to the sample actually measured, and an equation corresponding to the sample model having a periodic structure on the surface is calculated from equations (1) to (3) and (4) to (6) selected. The conditions are set as in the actual measurement, and an equation of an approximate sample model is selected. The surface microstructure analyzer 120 receives an input upon selection of an equation from the user (step S1). For the scattering intensity provided by the chosen equation, a shape factor F _j is an important element.

The X-rays entering the sample and propagating through a plurality of layers are diffracted and reflected not only by the sample surface but also by the interfaces between the layers (including the interface between the substrate and the film). With a larger number of layers, their influence increases. Consequently, using an equation in which the diffraction and the reflection at the interface are taken into account, it is possible to improve the accuracy of the analysis on the sample having a complicated surface structure.

Next, values necessary for calculation using the equation are received by input from the user and automatically transmitted by the X-ray diffraction meter 110 (step S2). As an initial fitting parameter, for example, values of the diameter of a circular cylinder, the height H, the top side of a trapezoid Wt, the length of the bottom side Wb, the height H, the radius of curvature of a convex end portion of the top side Rt, and the radius of curvature of a concave portion of the Base section provided. Furthermore, the total intensity is determined by the total number of structures N. As will be described later, by optimizing the values of each parameter [a, H, and NJ or [Wt, Wb, H, Rt, Rb, and NJ, it is possible to calculate a scattering intensity that agrees with the actually measured scattering intensity.

Subsequently, by using the equation as described above and the received values, the scattering intensity is calculated (step S3). The equation is calculated using the above parameters and thus the scattering intensity with respect to Qy and Qz on the detection surface is found.

Ii(exp)

Röntgenstreuintensität, welche an einem i-ten Messpunkt tatsächlich gemessen wurde

Ii(cal)

Röntgenstreuintensität, welche an dem i-ten Messpunkt berechnet wurde

Subsequently, the fit is performed on the calculated X-ray scattering intensity and the actually measured X-ray scattering intensity (step S4). Each of the X-ray scattering intensities is displayed as a curve on the detection surface. With this fit, the agreement (or a difference between both curves) of the curve that comes from the experiment is checked with the calculated curve. For example, the difference W between both curves is obtained by the following equation.
[Formula 7] $${\mathrm{<figure-callout\; id="2"\; label="W"\; filenames="DE112010001894B4\_0093.png,DE112010001894B4\_0094.png"\; state="\{\{state\}\}">W</figure-callout>}}^2={{\displaystyle \sum _i\left(\text{log}{I}_i\left(exp\right)-\text{log}{I}_i\left(cal\right)\right)}}^2$$

Ii(exp)

X-ray scattering intensity that was actually measured at an i-th measuring point

Ii(cal)

X-ray scattering intensity calculated at the i-th measurement point

Subsequently, if the difference W falls within a certain range, it is determined that both curves agree with each other, whereas if not, it is determined that both curves do not agree with each other (step S5).

If it is determined that the two curves do not match, the fitting parameters that determine the shape of the unit structures are changed (step S6), the X-ray scattering intensity is calculated again and a decision as to whether or not a match with the actual measured X-ray scattering intensity is carried out. This process is repeated while adjusting and changing the values of the fit parameters until both curves match each other.

When the calculated X-ray scattering intensity agrees with the actually measured X-ray scattering intensity, the selected values of the fit parameters are values that characterize the shape of the unit structure constituting the surface microstructure of the sample. The surface microstructure analyzer 120 outputs the results of the fitting parameters obtained (step S7) and completes the process. With this fit, for example, using a non-linear least squares method, it is possible to effectively determine the optimal value of each fit parameter. Although in the above example of fitting, the optimal value is calculated while adjusting the value of the fitting parameter, any fitting method can be used, and the fitting method is not particularly limited.

The calculation and fitting of the X-ray scattering intensity in the embodiment described above can be performed by the surface microstructure analyzer 120 using software which can be stored and executed by a computer. The surface microstructure analysis device 120 is preferably designed such that data can be exchanged between the X-ray scattering measurement device 110 and the surface microstructure analysis device 120 either bidirectionally or unidirectionally.

Preferably, when selecting the optimal values of the parameters by the simulation section 123 in order to improve the correspondence of the calculated X-ray scattering intensity with the actually measured X-ray scattering intensity (e.g. to be brought close to a certain value), the analysis is fully automatic by automatic selection by means of Apply a least squares method. The parameters can be entered freely and automatically. In each step, the calculation can be performed continuously and automatically, or can be performed by the user using a computer.

[Principle and derivation of equations]

(X-ray diffraction of a periodic array structure)

fµ

Streufaktor eines µ-Atoms

Xµ

Position des µ-Atoms

Q

Streuvektor

P

Polarisationsfaktor

rc

klassischer Elektronenradius (= 2,818 × 10^-15 m)

The derivation of equations used in the simulation described above is described. First, when the X-ray scattering/diffraction from the aggregation of the unit structure is considered, the basic equation for the scattering is as follows.
[Formula 8] $$I\left(Q\right)={\left|{\mathrm{right}}_{c}P{\displaystyle \sum _{\mu }{f}_{\mu }{\left(Q\right)}_e^{-iQ\cdot X_{\mu }}}\right|}^2={\left({\mathrm{right}}_{c}P\right)}^2{\displaystyle \sum _{\mu .v}{f}_{\mu }\left(Q\right)f_v^{*}{\left(Q\right)}_e^{-iQ\cdot \left(X_{\mu }-X_v\right)}}$$

fµ

Scattering factor of a µ-atom

Xµ

Position of the µ atom

Q

scatter vector

P

polarization factor

RC

classical electron radius (= 2.818 × 10 ^-15 m)

In the case of a small scattering angle, even if it is assumed that the atoms constituting the unit structure do not exist discretely but exist continuously, an excellent approximation can be obtained. Consequently, an internal coordinate r _u is introduced and the atomic position changes to X _µ → X _j + r _u . Here X _j is a position coordinate, which is typically of unit structure j. In this way, equation (7) can be rewritten as follows.
[Formula 9] $$\begin{array}{c}I\left(Q\right)={\left|{\mathrm{right}}_{c}P{\displaystyle \sum _j{e}^{-iQ\cdot X_j}{\displaystyle \underset{v}{\int }{\rho }_j\left({\mathrm{right}}_j\right)e^{-iQ\cdot {\mathrm{right}}_j}\mathrm{i.e}{v}_j}}\right|}^2\\ ={\left|{\mathrm{right}}_{c}P{\displaystyle \sum _j{f}_j\left(Q\right)e^{-iQ\cdot X_j}}\right|}^2={\left({\mathrm{right}}_{c}P\right)}^2{\displaystyle \sum _{j,k}{f}_j\left(Q\right)f_k^{*}\left(Q\right)e^{-iQ\cdot \left(X_j-X_k\right)}}\end{array}$$

Here, by assuming that the atoms are continuously distributed in the unit structure, the form factor is expressed as the following integral.
[Formula 10] $$f_j\left(Q\right)\equiv {\displaystyle \underset{v}{\int }{\rho }_j\left({\mathrm{right}}_j\right)e^{-iQ\cdot {\mathrm{right}}_j}}\mathrm{i.e}{v}_j$$

Then, assuming that the atoms are uniformly distributed in the unit structure, Equation (9) can be further simplified as the following equation.
[Formula 11] $$f_j^{\left(Q\right)}={\displaystyle \underset{v}{\int }{e}^{-iQ\cdot {\mathrm{right}}_j}}\mathrm{i.e}{v}_j$$

Since, as described above, the function characterizing the shape of the unit structure is derived, this function is also referred to as a shape factor or an external shape factor.

Xj

periodische Position

u(Xj)

Störung von der periodischen Position

Then, in order to determine the scattering intensity when the unit structures are arranged periodically, the sum of equation (8) is calculated considering the periodic structure and its “offset”. Displacement from the periodic structure in a crystal is formulated in terms of fluctuations in atomic positions due to thermal vibration, static displacement from the crystal lattice position of the atom due to crystal defects, and the like. This method is applied for the displacement from the periodic arrangement of the unit structures. Consequently, the position factor X _j of the unit structure is divided into an exact periodic position and an offset thereof.
[Formula 12] $$I\left(Q\right)={\left({\mathrm{right}}_c^P\right)}^2{\displaystyle \sum _{j,k}{f}_j\left(Q\right)f_k^{*}\left(Q\right)e^{-iQ\cdot \left(\mathrm{and}\left(X_j\right)-\mathrm{and}\left(X_k\right)\right)}{e}^{-iQ\cdot \left({\overline{X}}_j-{\overline{X}}_k\right)}}$$

Xj

periodic position

u(Xj)

Disturbance from the periodic position

Further, in order that the scattering intensity can be calculated considering fluctuations in the positions of each of the unit structures and fluctuations in the size, statistical processing is introduced.
[Formula 13] $$\begin{array}{c}I\left(Q\right)={\left({\mathrm{right}}_{c}P\right)}^2\left\{{\displaystyle \sum _j{\left|f_j^{\left(Q\right)}\right|}^2+{\displaystyle \sum _{j\ne k}{f}_j\left(Q\right)f_k^{*}\left(Q\right)e^{-iQ\cdot \left(\mathrm{and}\left(X_j\right)-\mathrm{and}\left(X_k\right)\right)}{e}^{-iQ\cdot \left({\overline{X}}_j-{\overline{X}}_k\right)}}}\right\}\\ ={\left({\mathrm{right}}_{c}P\right)}^{2}N\left\{{〈{\left|f_j\left(Q\right)\right|}^2〉}_j+{\displaystyle \sum _{\left|k\right|\ge 1}{〈f_j\left(Q\right)f_{k+j}^{*}\left(Q\right)〉}_j{〈e^{-iQ\cdot \left(\mathrm{and}\left(X_j\right)-\mathrm{and}\left(X_k\right)\right)}〉}_j{e}^{-iQ\left({\overline{X}}_j-{\overline{X}}_{k+j}\right)}}\right\}\end{array}$$

The following operator characterizes the statistical mean of a physical observable A with respect to the total number of unit structures N.
[Formula 14] $${〈A_j〉}_j\equiv \frac{1}{N}{\displaystyle \sum _j{A}_j}$$

Here, due to the uniformity of the whole system, it is assumed that a correlation function depends only on the relative positional relation k of the two.

Next, an exponential function is first expanded taking into account fluctuations in the positions. For simplicity, assume that the following relation is true: $$\left|Q\cdot \left(\mathrm{and}\left(X_j\right)\right)-\mathrm{and}\left(X_{j+k}\right)\right|<<1$$

Then the following formula is obtained.
[Formula 16] $${〈e^{-iQ\cdot \left(\mathrm{and}\left(X_j\right)-\mathrm{and}\left(X_{j+k}\right)\right)}〉}_j\approx 1+{〈-iQ\cdot \left(\mathrm{and}\left(X_j\right)-\mathrm{and}\left(X_{j+k}\right)\right)〉}_j+\frac{1}{2}{〈-iQ\cdot {\left(\mathrm{and}\left(X_j\right)-\mathrm{and}\left(X_{j+k})\right)\right)}^2〉}_j+\text{L}$$

Since there are certainly positive and negative terms in the relation of relative positions, these are averaged and thus the first term becomes zero and the second term remains. Consequently, the relation of the relative positions can be expressed as the following formula in the range of this approximation.
[Formula 17] $$\begin{array}{c}{〈e^{-iQ\cdot \left(\mathrm{and}\left(X_j\right)-\mathrm{and}\left(X_{j+k}\right)\right)}〉}_j\approx e^{-\frac{{〈{\left(Q\cdot \left(\mathrm{and}\left(X_j\right)-\mathrm{and}\left(X_{j+k}\right)\right)\right)}^2〉}_{\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="DE112010001894B4\_0093.png,DE112010001894B4\_0094.png"\; state="\{\{state\}\}">j</figure-callout>}}}{2}}+\text{L}\\ =e^{-\frac{Q_{\mathrm{and}}^2〈\mathrm{\Delta }\mathrm{and}{\left(\mathrm{\Delta }{X}_k\right)}^2〉}{2}}+\text{L}\\ =e^{-\frac{Q_{\mathrm{and}}^{2}G\left(\mathrm{\Delta }{X}_k\right)}{2}}+\text{L}\end{array}$$

g(ΔXk)

Funktion in Abhängigkeit der relativen Position

However, it is assumed that the correlation of the displacement is expressed as a function g(ΔX _k ) of only the difference ΔX _k between respective positions. Further, Q _u represents the projection of a scattering vector Q in the u direction of the dislocation. The correlation of the fluctuation with respect to the positions causes so-called Hung scattering in the X-ray scattering. As a simple example of the correlation, specific expressions are given for a case where random displacement occurs regardless of the distance and a case where the position fluctuates periodically. Equation (6) can be written as follows using g(ΔX _k ).
[Formula 18] $$I\left(Q\right)={〈{\mathrm{right}}_{c}P〉}^{2}N\left\{{〈{\left|f_j\left(Q\right)\right|}^2〉}_j+{\displaystyle \sum _{\left|k\right|\ge 1}{〈f_j\left(Q\right)f_{j+k}^{*}\left(Q\right)〉}_j{e}^{-iQ\cdot \left({\overline{X}}_j-{\overline{X}}_{j+k}\right)}{e}^{-\frac{Q_{\mathrm{and}}^{2}G\left(\mathrm{\Delta }{X}_k\right)}{2}}}\right\}$$

g(ΔXk)

Function depending on the relative position

$$g_{\left(\mathrm{\Delta }{X}_k\right)}=b^2{\left(1-\text{cos}\left(2\pi \frac{\left|\mathrm{\Delta }{X}_k\right|}{p}\right)\right)}^2$$

When g(ΔX _k ) is unrelated to distance, as shown in equation (16a), this is expressed as a constant mean square fluctuation. The mean square fluctuations, which are expressed independently of distance, behave like a temperature factor in crystal diffraction. If the amplitude of the fluctuations with respect to the position having a period p is assumed to be b, this can be expressed by the following equation (16b).
[Formula 19] $$G_{\left(\mathrm{\Delta }{X}_k\right)}=\mathrm{\Delta }{\overline{\mathrm{and}}}^2$$

$$G_{\left(\mathrm{\Delta }{X}_k\right)}=b^2{\left(1-\text{cos}\left(2\pi \frac{\left|\mathrm{\Delta }{X}_k\right|}{p}\right)\right)}^2$$

Then, a diffraction intensity can be expressed as follows.
[Formula 20] $$I\left(Q\right)={\left({\mathrm{right}}_{c}P\right)}^{2}N\left\{{〈{\left|f_j\left(Q\right)\right|}^2〉}_j+{\displaystyle \sum _{\left|k\right|\ge 1}{〈f_j\left(Q\right)f_{j+k}^{*}\left(Q\right)〉}_j{e}^{-iQ\left({\overline{X}}_j-{\overline{X}}_{j+k}\right)}{e}^{-\frac{Q_{\mathrm{and}}^2}{2}{b}^2{\left(1-\text{cos}\left(2\pi \frac{\mathrm{\Delta }{X}_k}{p}\right)\right)}^2}}\right\}$$

With this equation, it is possible to express not only a diffraction peak resulting from an original period but also a peak resulting from a superlattice period p times as high.

In general, the structure fluctuation of the unit structure in which the shape factor is expressed as Equation (9) or (10) can have a correlation depending on the difference ΔX _k between the corresponding or mutual positions. However, if such a correlation can be ignored, the shape factor correlation can be expressed by the following equation. $${〈f_j\left(Q\right)f_{j+k}^{*}\left(Q\right)〉}_j={〈f_j\left(Q\right)〉}_j{〈f_{j+k}^{*}\left(Q\right)〉}_j={\left|〈f\left(Q\right)〉\right|}^2$$

$$I\left(Q\right)={\left(r_{c}P\right)}^{2}N\left\{\langle {\left|F\left(Q\right)\right|}^2\rangle +{\left|\langle F\left(Q\right)\rangle \right|}^2{e}^{-\frac{Q_u^2{\mathrm{\Delta }}_u^{-2}}{2}}{\displaystyle \sum _{\left|k\right|\ge 1}{e}^{-iQ\cdot \left({\overline{X}}_j-{\overline{X}}_{j+k}\right)}}\right\}$$

$$I\left(Q\right)={\left(r_{c}P\right)}^{2}N\left\{\langle {\left|F^{\left(Q\right)}\right|}^2\rangle +{\left|\langle F^{\left(Q\right)}\rangle \right|}^2{\displaystyle \sum _{\left|k\right|\ge 1}{}_e^{-iQ\cdot \left({\overline{X}}_j-{\overline{X}}_{j+k}\right)}{e}^{-\frac{Q_u^2}{2}{b}^2{\left(1-\text{cos}\left(2\pi \frac{\left|\mathrm{\Delta }{X}_k\right|}{p}\right)\right)}^2}}\right\}$$

Then, Equations (12), (16) and (17) can be simply expressed as follows.
[Formula 22] $$I\left(Q\right)={\left({\mathrm{right}}_{c}P\right)}^{2}N\left\{〈{\left|f\left(Q\right)\right|}^2〉+{\left|〈f\left(Q\right)〉\right|}^2{\displaystyle \sum _{\left|k\right|\ge 1}{〈e^{-iQ\cdot \left(\mathrm{and}\left(X_j\right)-\mathrm{and}\left(X_k\right)\right)}〉}_j{e}^{-iQ\cdot \left({\overline{X}}_j-{\overline{X}}_{j+k}\right)}}\right\}$$

$$I\left(Q\right)={\left({\mathrm{right}}_{c}P\right)}^{2}N\left\{〈{\left|f\left(Q\right)\right|}^2〉+{\left|〈f\left(Q\right)〉\right|}^2{e}^{-\frac{Q_{\mathrm{and}}^2{\mathrm{\Delta }}_{\mathrm{and}}^{-2}}{2}}{\displaystyle \sum _{\left|k\right|\ge 1}{e}^{-iQ\cdot \left({\overline{X}}_j-{\overline{X}}_{j+k}\right)}}\right\}$$

$$I\left(Q\right)={\left({\mathrm{right}}_{c}P\right)}^{2}N\left\{〈{\left|f^{\left(Q\right)}\right|}^2〉+{\left|〈f^{\left(Q\right)}〉\right|}^2{\displaystyle \sum _{\left|k\right|\ge 1}{}_e^{-iQ\cdot \left({\overline{X}}_j-{\overline{X}}_{j+k}\right)}{e}^{-\frac{Q_{\mathrm{and}}^2}{2}{b}^2{\left(1-\text{cos}\left(2\pi \frac{\left|\mathrm{\Delta }{X}_k\right|}{p}\right)\right)}^2}}\right\}$$

It should be noted that since in the equation of the form factor of the unit structure, the first term is the square of itself, the averaging is performed after squaring, and that since the subsequent term is a term obtained as a result of interference between different scattering elements occurs, the averaging is performed on each form factor, and then the squaring is performed. Consequently, when the periodic structure is continuous over a long distance, the second term has a significant contribution, whereas when the regularity is small, the contribution of the second term changes gradually according to the magnitude of Q. Applying these equations, it is possible to adequately describe how the method of averaging the shape factors changes. However, in general, the coherence range of X-rays is narrow and it is unlikely that all of the observed ranges coherently contribute to the scattering. In such a case, in order to take the distribution of the scatter into account, it is necessary to carry out a new averaging according to the distribution of each of the above three equations.

(X-ray diffraction considering reflection/diffraction at surface or thin film)

The purpose of the present invention is to carry out a structure analysis on unit structures on the surface. Consequently, the X-rays hit the surface at a small angle near the surface and the diffraction/scattering is measured. In such a case it is necessary to carry out a diffraction intensity calculation taking into account reflection/diffraction effects of the X-rays at the surface. The calculation procedure is described below.

When the X-rays hit the sample surface at a small angle, reflection and diffraction at the surface and interfaces are very important. When measuring small-angle scattering of the reflected X-ray, it is necessary to take this into account. The 7 and 8th 12 are schematic diagrams showing the state of an electric field in the layers of the sample, which generally have an N-layer multilayer structure. Here, ^T E _m and ^R E _m denote a propagating wave and a reflected wave, respectively, within m layers. These values can be calculated based on the Fresnel formula given the refractive index n _m and the thickness d _m of each layer and the angle of incidence of X-rays α _o .

reflektierte Welle und fortpflanzende Welle in einer gestreuten Welle wobei
$T_{E_0^{*}}^{~}:$

Welle, die von der Oberfläche emittiert wirdRegarding a scattered wave, it is necessary to consider a wave which is generated in the film and which is emitted from the surface at an exit angle a ₀ . As a solution of a wave equation representing an electric field in the multilayer film that satisfies such conditions, a solution obtained by time reversal of a normal solution can be used. This can be obtained by obtaining the complex conjugate of a normal solution and then changing it so that t → -t (k → -k). This solution is represented by the following icons.
[Formula 23] $T_{E_m^{*}}^{~},R_{E_m^{*}}^{~}:$

reflected wave and propagating wave being in a scattered wave
${\mathrm{<figure-callout\; id="0"\; label="T\; E"\; filenames="DE112010001894B4\_0075.png,DE112010001894B4\_0086.png"\; state="\{\{state\}\}">T</figure-callout>}}_{E_{}^{}}^{{}_0^{*}}:$

Wave emitted from the surface

$${\eta }_m=\sqrt{n_m-co{s}^2{\alpha }_0},{\gamma }_m=\frac{{\eta }_m-{\eta }_{m+1}}{{\eta }_m+{\eta }_{m+1}},{\tau }_m=\frac{2{\eta }_{m+1}}{{\eta }_m+{\eta }_{m+1}},{\phi }_m=e^{i{k}_0{\eta }_m,d_m},t_m=\frac{1-{\gamma }_m{R}_m}{{\tau }_m}$$

$${}^{T}E{}_m\left(z_m\right)={\displaystyle \prod _{j=0}^{m-1}\left(t_j{\phi }_j\right)e^{-i{k}_0{\eta }_m{z}_m}}=T_m{e}^{-i{k}_0{\eta }_m{z}_m},{}^R{E}_m\left(z_m\right)={\displaystyle \prod _{j=0}^{m-1}{R}_m\left(t_j{\phi }_j\right)e^{i{k}_0{\eta }_m{z}_m}}=T_m{R}_m{e}^{i{k}_0{\eta }_m{z}_m}$$

Specifically, the electric field resulting from the incident wave (wave 1) can be described as follows.
[Formula 24] $$R_N=\mathrm{0,}{R}_{N-1}=g_{N-1},\text{L}\text{,}{R}_m=\frac{R_{m+1}{\phi }_{m+1}^2+g_m}{R_{m+1}{\phi }_{m+1}^2{g}_{m+1}},{\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="DE112010001894B4\_0075.png,DE112010001894B4\_0086.png"\; state="\{\{state\}\}">R</figure-callout>}}_0=\frac{{\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="DE112010001894B4\_0074.png,DE112010001894B4\_0075.png"\; state="\{\{state\}\}">R</figure-callout>}}_1{\mathrm{<figure-callout\; id="1"\; label="\phi "\; filenames="DE112010001894B4\_0074.png,DE112010001894B4\_0075.png"\; state="\{\{state\}\}">\phi </figure-callout>}}_1^2+g_0}{{\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="DE112010001894B4\_0074.png,DE112010001894B4\_0075.png"\; state="\{\{state\}\}">R</figure-callout>}}_1{\mathrm{<figure-callout\; id="1"\; label="\phi "\; filenames="DE112010001894B4\_0074.png,DE112010001894B4\_0075.png"\; state="\{\{state\}\}">\phi </figure-callout>}}_1^2{\mathrm{<figure-callout\; id="0"\; label="g"\; filenames="DE112010001894B4\_0075.png,DE112010001894B4\_0086.png"\; state="\{\{state\}\}">g</figure-callout>}}_0+1}$$

$$n_m=\sqrt{n_m-cO{s}^2{a}_0},g_m=\frac{n_m-n_{m+1}}{n_m+n_{m+1}},{\tau }_m=\frac{2n_{m+1}}{n_m+n_{m+1}},{\phi }_m=e^{i{k}_0{n}_m,{\mathrm{i.e}}_m},t_m=\frac{1-g_m{R}_m}{{\tau }_m}$$

$${}^{T}E{}_m\left({\mathrm{e.g}}_m\right)={\displaystyle \prod _{j=0}^{m-1}\left(t_j{\phi }_j\right)e^{-i{k}_0{n}_m{\mathrm{e.g}}_m}}=T_m{e}^{-i{k}_0{n}_m{\mathrm{e.g}}_m},{}^R{E}_m\left({\mathrm{e.g}}_m\right)={\displaystyle \prod _{j=0}^{m-1}{R}_m\left(t_j{\phi }_j\right)e^{i{k}_0{n}_m{\mathrm{e.g}}_m}}=T_m{R}_m{e}^{i{k}_0{n}_m{\mathrm{e.g}}_m}$$

$${\zeta }_m=\sqrt{n_m-co{s}^2{\beta }_0},{\tilde{\gamma }}_m^{*}=\frac{{\zeta }_m^{*}-{\zeta }_{m+1}^{*}}{{\zeta }_m^{*}-{\zeta }_{m+1}^{*}},{\tilde{\tau }}_m^{*}=\frac{2{\zeta }_{m+1}^{*}}{{\zeta }_m^{*}+{\zeta }_{m+1}^{*}},{\tilde{\phi }}_m^{*}=e^{-i{k}_0{\zeta }_m^{*}{d}_m},{\tilde{t}}_m^{*}=\frac{1-{\tilde{\gamma }}_m^{*}{\tilde{R}}_m^{*}}{{\tilde{\tau }}_m^{*}}$$

$${}^T{\tilde{E}}_m^{*}\left(z_m\right)={\displaystyle \prod _{j=0}^{m-1}\left({\tilde{t}}_j^{*}{\tilde{\phi }}_j^{*}\right)e^{i{k}_0{\zeta }_m^{*}{z}_m}={\tilde{T}}_m^{*}{e}^{i{k}_0{\zeta }_m^{*}{z}_m},{}^R{\tilde{E}}_m^{*}\left(z_m\right)={\displaystyle \prod _{j=0}^{m-1}{R}_m^{*}\left({\tilde{t}}_j^{*}{\tilde{\phi }}_m^{*}\right)e^{-i{k}_0{\zeta }_m^{*}{z}_m}={\tilde{T}}_m^{*}{R}_m^{*}{e}^{-i{k}_0{\zeta }_m^{*}{z}_m}}}$$

The scattered wave (wave 2) is likewise given by the following formula.
[Formula 25] $${\tilde{R}}_N^{*}=\mathrm{0,}{\tilde{R}}_{N-1}^{*}={\tilde{g}}_{N-1}^{*},\text{L}\text{,}{\tilde{R}}_I^{*}=\frac{{\tilde{R}}_{m+1}^{*}{\tilde{\phi }}_{m+1}^{{*}^2}+{\tilde{g}}_m^{*}}{{\tilde{R}}_{m+1}^{*}{\phi }_{m+1}^{{*}^2}{\tilde{g}}_m^{*}+1},{\tilde{R}}_0^{*}=\frac{{\tilde{R}}_1^{*}{\tilde{\phi }}_1^{{*}^2}+{\tilde{g}}_0^{*}}{{\tilde{R}}_1^{*}{\tilde{\phi }}_1^{{*}^2}{\tilde{g}}_0^{*}+1}$$

$${\zeta }_m=\sqrt{n_m-cO{s}^2{\mathrm{<figure-callout\; id="0"\; label="\beta "\; filenames="DE112010001894B4\_0075.png,DE112010001894B4\_0086.png"\; state="\{\{state\}\}">\beta </figure-callout>}}_0},{\tilde{g}}_m^{*}=\frac{{\zeta }_m^{*}-{\zeta }_{m+1}^{*}}{{\zeta }_m^{*}-{\zeta }_{m+1}^{*}},{\tilde{\tau }}_m^{*}=\frac{2{\zeta }_{m+1}^{*}}{{\zeta }_m^{*}+{\zeta }_{m+1}^{*}},{\tilde{\phi }}_m^{*}=e^{-i{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="DE112010001894B4\_0075.png,DE112010001894B4\_0086.png"\; state="\{\{state\}\}">k</figure-callout>}}_0{\zeta }_m^{*}{\mathrm{i.e}}_m},{\tilde{t}}_m^{*}=\frac{1-{\tilde{g}}_m^{*}{\tilde{R}}_m^{*}}{{\tilde{\tau }}_m^{*}}$$

$${}^T{\tilde{E}}_m^{*}\left({\mathrm{e.g}}_m\right)={\displaystyle \prod _{j=0}^{m-1}\left({\tilde{t}}_j^{*}{\tilde{\phi }}_j^{*}\right)e^{i{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="DE112010001894B4\_0075.png,DE112010001894B4\_0086.png"\; state="\{\{state\}\}">k</figure-callout>}}_0{\zeta }_m^{*}{\mathrm{e.g}}_m}={\tilde{T}}_m^{*}{e}^{i{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="DE112010001894B4\_0075.png,DE112010001894B4\_0086.png"\; state="\{\{state\}\}">k</figure-callout>}}_0{\zeta }_m^{*}{\mathrm{e.g}}_m},{}^R{\tilde{E}}_m^{*}\left({\mathrm{e.g}}_m\right)={\displaystyle \prod _{j=0}^{m-1}{R}_m^{*}\left({\tilde{t}}_j^{*}{\tilde{\phi }}_m^{*}\right)e^{-i{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="DE112010001894B4\_0075.png,DE112010001894B4\_0086.png"\; state="\{\{state\}\}">k</figure-callout>}}_0{\zeta }_m^{*}{\mathrm{e.g}}_m}={\tilde{T}}_m^{*}{R}_m^{*}{e}^{-i{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="DE112010001894B4\_0075.png,DE112010001894B4\_0086.png"\; state="\{\{state\}\}">k</figure-callout>}}_0{\zeta }_m^{*}{\mathrm{e.g}}_m}}}$$

These quantities can be calculated given the parameters n _m and d _m of the angle of incidence, the angle of exit and the film structure. Instead of the above formula, the following formula can also be used as a formula considering the interface roughness Ω _m .
[Formula 26] $$g_m=\frac{n_m-n_{m+1}}{n_m+n_{m+1}}\text{ex}\left[-2{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="DE112010001894B4\_0075.png,DE112010001894B4\_0086.png"\; state="\{\{state\}\}">k</figure-callout>}}_0^2{n}_m{n}_{m+1}{\sigma }_m^2\right],{\tilde{g}}_m^{*}=\frac{{\zeta }_m^{*}-{\zeta }_{m+1}^{*}}{{\zeta }_m^{*}+{\zeta }_{m+1}^{*}}\text{ex}\left[-2{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="DE112010001894B4\_0075.png,DE112010001894B4\_0086.png"\; state="\{\{state\}\}">k</figure-callout>}}_0^2{\zeta }_m^{*}{\zeta }_{m+1}^{*}{\sigma }_m^2\right]$$

$${\tilde{\psi }}_m^f\left(\beta \right)={\displaystyle \prod _{j=1}^{m-1}\left({\tilde{t}}_j^{*}{\tilde{\phi }}_j^{*}\right)e^{-i{k}_0{\zeta }_m^{*}{z}_m}+{\displaystyle \prod _{j=1}^{m-1}{\tilde{R}}_m^{*}\left({\tilde{t}}_j^{*}{\tilde{\phi }}_j^{*}\right)}{e}^{-i{k}_0{\zeta }_m^{*}{z}_m}={\tilde{T}}_m^{*}{e}^{-i{k}_0{\zeta }_m^{*}{z}_m}+{\tilde{T}}_m^{*}{\tilde{R}}_m^{*}{e}^{-i{k}_0{\zeta }_m^{*}{z}_m}}$$

$$\begin{array}{c}\langle {\tilde{\psi }}_m^f\left(\beta \right)\left|V\right|{\psi }^i\left(\alpha \right)\rangle ={\displaystyle \sum _m\langle {\tilde{\psi }}_m^f\left(\beta \right)\left|V_m\right|{\psi }_m^i\left(\alpha \right)\rangle }\\ ={\displaystyle \sum _m{\tilde{T}}_m{T}_m\left(\langle {\zeta }_m^{*}\left|V_m\right|{\eta }_m\rangle +R_m\langle {\zeta }_m^{*}\left|V_m\right|-{\eta }_m\rangle +{\tilde{R}}_m\langle -{\zeta }_m^{*}\left|V_m\right|{\eta }_m\rangle +{\tilde{R}}_m{R}_m\langle -{\zeta }_m^{*}\left|V_m\right|-{\eta }_m\rangle \right)}\end{array}$$

Using these, a transition amplitude from wave 1 to wave 2 caused by a potential V _m due to uneven density in the m layers can be written as follows become.
[Formula 27] $${\psi }_l^i\left(a\right)={\displaystyle \prod _{j=1}^{l-1}\left(t_j{\phi }_j\right)e^{-i{k}_0{n}_m{\mathrm{e.g}}_m}}+{\displaystyle \prod _{j=1}^{m-1}{R}_m\left(t_j{\phi }_j\right)e^{-i{k}_0{n}_m{\mathrm{e.g}}_m}=T_m{e}^{-i{k}_0{n}_m{\mathrm{e.g}}_m}+T_m{R}_m{e}^{-i{k}_0{n}_m{\mathrm{e.g}}_m}}$$

$${\tilde{\psi }}_m^f\left(\beta \right)={\displaystyle \prod _{j=1}^{m-1}\left({\tilde{t}}_j^{*}{\tilde{\phi }}_j^{*}\right)e^{-i{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="DE112010001894B4\_0075.png,DE112010001894B4\_0086.png"\; state="\{\{state\}\}">k</figure-callout>}}_0{\zeta }_m^{*}{\mathrm{e.g}}_m}+{\displaystyle \prod _{j=1}^{m-1}{\tilde{R}}_m^{*}\left({\tilde{t}}_j^{*}{\tilde{\phi }}_j^{*}\right)}{e}^{-i{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="DE112010001894B4\_0075.png,DE112010001894B4\_0086.png"\; state="\{\{state\}\}">k</figure-callout>}}_0{\zeta }_m^{*}{\mathrm{e.g}}_m}={\tilde{T}}_m^{*}{e}^{-i{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="DE112010001894B4\_0075.png,DE112010001894B4\_0086.png"\; state="\{\{state\}\}">k</figure-callout>}}_0{\zeta }_m^{*}{\mathrm{e.g}}_m}+{\tilde{T}}_m^{*}{\tilde{R}}_m^{*}{e}^{-i{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="DE112010001894B4\_0075.png,DE112010001894B4\_0086.png"\; state="\{\{state\}\}">k</figure-callout>}}_0{\zeta }_m^{*}{\mathrm{e.g}}_m}}$$

$$\begin{array}{c}〈{\tilde{\psi }}_m^f\left(\beta \right)\left|V\right|{\psi }^i\left(a\right)〉={\displaystyle \sum _m〈{\tilde{\psi }}_m^f\left(\beta \right)\left|V_m\right|{\psi }_m^i\left(a\right)〉}\\ ={\displaystyle \sum _m{\tilde{T}}_m{T}_m\left(〈{\zeta }_m^{*}\left|V_m\right|n_m〉+R_m〈{\zeta }_m^{*}\left|V_m\right|-n_m〉+{\tilde{R}}_m〈-{\zeta }_m^{*}\left|V_m\right|n_m〉+{\tilde{R}}_m{R}_m〈-{\zeta }_m^{*}\left|V_m\right|-n_m〉\right)}\end{array}$$

The square of the absolute value of Equation 27 obtained here gives the scatter probability. The evaluation of surface scattering described above is called "distorted wave born approximation" (DWBA).

(Form (structure) factor of a surface nanostructure)

Based on the basic calculations described above, the structure factor of an actual nanostructure is given and the x-ray scattering intensity is specified. 9 12 is a cross-sectional view of a sample model in which lines and spaces are formed on the surface thereof. When the X-ray hits the unit structures arranged at intervals sufficiently smaller than a coherence length and present on the surface at a grazing angle near a critical angle, the X-ray becomes not only on the bottom surface but also on the top surface reflected in the unit structure. Then, an interface pattern reflecting the height H of the unit structure appears on a reflection pattern of the X-ray as if a thin film were formed on the surface.

wobei θ(x) eine Stufenfunktion ist, die wie folgt ausgedrückt wird.
[Formel 29] $$\theta \left(x\right)=\{\begin{array}{cc}1& x>0\\ 0& x<0\end{array}$$

Taking this into account, regarding, for example, a structure formed on the surface with a high density, it has been found that it is necessary to calculate a surface electromagnetic field on a layered structure having a periodic structure in each layer, and to deal with the problem of scattering from the surface structure based on this. 10 12 is a cross-sectional view showing a sample model in which a layered structure is formed. A reflected wave and a surface reflected/diffracted wave will be described later, and a scattering amplitude when there is a scattering source having the potential V in the film represented by Equations (25 to 27) will be described. For example, in order for the scattering amplitude of the surface shape used in 9 shown can be calculated, the surface shape is first represented as the height Z(X,Y) of the surface at each point (X,Y). Although in the example of 9 the unit structures are present only in the first layer, the unit structures may be present substantially over a plurality of layers. When the unit structures are present over a plurality of layers on the surface, the scattering amplitude at the m-th layer (Lm) is expressed as follows.
[Formula 28] $$〈{\tilde{\psi }}_m^f\left(\beta \right)\left|V_m\right|{\psi }_m^f\left(a\right)〉={\mathrm{right}}_{c}P{\tilde{T}}_m{T}_m\left[\begin{array}{l}{\displaystyle \underset{0}{\overset{{\mathrm{i.e}}_m}{\int }}\theta \left(Z_m\left(X,Y\right)-{\mathrm{e.g}}_m\right)e^{-i{k}_0\left(n_m-{\zeta }_m\right){\mathrm{e.g}}_m}\mathrm{i.e}{\mathrm{e.g}}_m{\displaystyle \underset{\infty }{\overset{\infty }{\int }}{e}^{-i{Q}_{y}Y}}\mathrm{i.e}Y{\displaystyle \underset{\infty }{\overset{\infty }{\int }}{e}^{-i{Q}_{x}X}}\mathrm{i.e}X}\hfill \\ +{\tilde{R}}_m{\displaystyle \underset{0}{\overset{{\mathrm{i.e}}_m}{\int }}\theta \left(Z_m\left(X,Y\right)-{\mathrm{e.g}}_m\right)e^{-i{k}_0\left(n_m-{\zeta }_m\right){\mathrm{e.g}}_m}\mathrm{i.e}{\mathrm{e.g}}_m{\displaystyle \underset{\infty }{\overset{\infty }{\int }}{e}^{-i{Q}_{y}Y}\mathrm{i.e}Y}{\displaystyle \underset{\infty }{\overset{\infty }{\int }}{e}^{-i{Q}_{x}X}\mathrm{i.e}X}}\hfill \\ +R_m{\displaystyle \underset{0}{\overset{{\mathrm{i.e}}_m}{\int }}\theta \left(Z_m\left(X,Y\right)-{\mathrm{e.g}}_m\right)e^{-i{k}_0\left(-n_m+{\zeta }_m\right){\mathrm{e.g}}_m}\mathrm{i.e}{\mathrm{e.g}}_m{\displaystyle \underset{\infty }{\overset{\infty }{\int }}{e}^{-i{Q}_{y}Y}\mathrm{i.e}Y{\displaystyle \underset{\infty }{\overset{\infty }{\int }}{e}^{-i{Q}_{x}X}\mathrm{i.e}X}}}\hfill \\ {\tilde{R}}_m{R}_m{\displaystyle \underset{0}{\overset{{\mathrm{i.e}}_m}{\int }}\theta \left(Z_m\left(X,Y\right)-{\mathrm{e.g}}_m\right)e^{-i{k}_0\left(-n_m-{\zeta }_m\right){\mathrm{e.g}}_m}\mathrm{i.e}{\mathrm{e.g}}_m{\displaystyle \underset{\infty }{\overset{\infty }{\int }}{e}^{-i{Q}_{y}Y}\mathrm{i.e}Y{\displaystyle \underset{\infty }{\overset{\infty }{\int }}{e}^{-i{Q}_{x}X}\mathrm{i.e}X}}}\hfill \end{array}\right]$$

where θ(x) is a step function expressed as follows.
[Formula 29] $$\theta \left(x\right)=\{\begin{array}{cc}1& x>0\\ 0& x<0\end{array}$$

Here, when 0< _Zm (X,Y)< _dm , the integration of _Zm in equation (28) can be performed as follows.
[Formula 30] $$\begin{array}{c}{\displaystyle \underset{0}{\overset{{\mathrm{i.e}}_m}{\int }}\theta \left(Z_m\left(X,Y\right)-{\mathrm{e.g}}_m\right)e^{-i{k}_0\left(n_m+{\zeta }_m\right){\mathrm{e.g}}_m}\mathrm{i.e}{\mathrm{e.g}}_m}\\ ={\left[\frac{\theta \left(Z_m\left(X,Y\right)-{\mathrm{e.g}}_m\right)e^{-i{k}_0\left(n_m+{\zeta }_m\right){\mathrm{e.g}}_m}}{-i{k}_0\left(n_m+{\zeta }_m\right)}\right]}_0^{{\mathrm{i.e}}_m}+{\displaystyle \underset{0}{\overset{{\mathrm{i.e}}_m}{\int }}\frac{\delta \left(Z_m\left(X,Y\right)-{\mathrm{e.g}}_m\right)e^{-i{k}_0\left(n_m+{\zeta }_m\right){\mathrm{e.g}}_m}}{-i{k}_0\left(n_m+{\zeta }_m\right)}}\mathrm{i.e}{\mathrm{e.g}}_m\\ =\frac{\theta \left(Z_m\left(X,Y\right)-{\mathrm{e.g}}_m\right)e^{-i{k}_0\left(n_m+{\zeta }_m\right){\mathrm{e.g}}_m}-\theta \left(Z_m\left(X,Y\right)-0\right)}{-i{k}_0\left(n_m+{\zeta }_m\right)}+\frac{e^{-i{k}_0\left(n_m+{\zeta }_m\right)Z_m\left(X,Y\right)}}{-i{k}_0\left(n_m+{\zeta }_m\right)}\\ =\frac{e^{-i{k}_0\left(n_m+{\zeta }_m\right)Z_m\left(X,Y\right)}-1}{-i{k}_0\left(n_m+{\zeta }_m\right)}\end{array}$$

Then it is assumed that each term is expressed with the following equations.
[Formula 31] $$\{\begin{array}{c}{G}_m^{T\tilde{T}}\left(n_m,{\zeta }_m,Q_x,Q_y\right)={\displaystyle \underset{s}{\int }\frac{e^{-i{k}_0\left(n_m+{\zeta }_m\right)Z_m\left(X,Y\right)}-1}{-i{k}_0\left(n_m+{\zeta }_m\right)}{e}^{-i\left(Q_x\cdot X+Q_y\cdot Y\right)}\mathrm{i.e}X\mathrm{i.e}Y}\\ G_m^{R\tilde{T}}\left(n_m,{\zeta }_m,Q_x,Q_y\right)={\displaystyle \underset{s}{\int }\frac{e^{-i{k}_0\left(-n_m+{\zeta }_m\right)Z_m\left(X,Y\right)}-1}{-i{k}_0\left(-n_m+{\zeta }_m\right)}{e}^{-i\left(Q_x\cdot X+Q_y\cdot Y\right)}\mathrm{i.e}X\mathrm{i.e}Y}\\ G_m^{T\tilde{R}}\left(n_m,{\zeta }_m,Q_x,Q_y\right)={\displaystyle \underset{s}{\int }\frac{e^{-i{k}_0\left(n_m-{\zeta }_m\right)Z_m\left(X,Y\right)}-1}{-i{k}_0\left(n_m-{\zeta }_m\right)}{e}^{-i\left(Q_x\cdot X+Q_y\cdot Y\right)}\mathrm{i.e}X\mathrm{i.e}Y}\\ G_m^{R\tilde{R}}\left(n_m,{\zeta }_m,Q_x,Q_y\right)={\displaystyle \underset{s}{\int }\frac{e^{-i{k}_0\left(-n_m-{\zeta }_m\right)Z_m\left(X,Y\right)}-1}{-i{k}_0\left(-n_m-{\zeta }_m\right)}{e}^{-i\left(Q_x\cdot X+Q_y\cdot Y\right)}\mathrm{i.e}X\mathrm{i.e}Y}\end{array}$$

Consequently, Equation (27) on scattering amplitude can be expressed as follows.
[Formula 32] $$〈{\tilde{\psi }}^f\left(\beta \right)\left|V\right|{\psi }^i\left(a\right)〉={\mathrm{right}}_{c}P{\displaystyle \sum _m{T}_m{\tilde{T}}_m\left\{G_m^{T\tilde{T}}+R_m\cdot G_m^{R\tilde{T}}+{\tilde{R}}_m\cdot G_m^{T\tilde{R}}+R_m{\tilde{R}}_m\cdot G_m^{R\tilde{R}}\right\}}$$

$$\begin{array}{l}{Q}_z^{T_m{\tilde{T}}_m}=k_0\left({\eta }_m+{\zeta }_m\right)\hfill \\ Q_z^{R_m{\tilde{T}}_m}=k_0\left(-{\eta }_m+{\zeta }_m\right)\hfill \\ Q_z^{T_m{\tilde{R}}_m}=k_0\left({\eta }_m-{\zeta }_m\right)\hfill \\ Q_z^{R_m{\tilde{R}}_m}=k_0\left(-{\eta }_m-{\zeta }_m\right)\hfill \end{array}\}\{\begin{array}{c}{\eta }_m=\sqrt{n_m-{\text{cos}}^2\alpha }\\ {\zeta }_m=\sqrt{n_m-{\text{cos}}^2\beta }\end{array}$$

Furthermore, a case where the unit structures have a periodicity in the y-direction is considered. Such a case has been discussed with reference to equations (7) to (17'). This is applied to a specific calculation of equation (32). For example, to generally deal with the surface nanostructure, consider a case where a two-dimensional periodic structure is formed in the surface. Hence, here the point (X, Y) within the unit structures constituting the periodic structure is given as (X, Y) = (Xj + xj, YJ + yj) using the local coordinate (X _j , Y _j ) and the expressing the position coordinate (X _j , Y _j ) of each cell, and rewriting the equation (31), with the result that the following formula is given.
[Formula 33] $$\begin{array}{c}{G}_m\left(Q_{\mathrm{e.g}}^{am},Q_x,Q_y\right)={\displaystyle \underset{S}{\int }\frac{e^{-i{Q}_{\mathrm{e.g}}^{am}{Z}_m\left(X,Y\right)}-1}{-i{Q}_{\mathrm{e.g}}^{am}}}{e}^{-i\left(Q_x\cdot X+Q_y\cdot Y\right)}\mathrm{i.e}X\mathrm{i.e}Y\\ ={\displaystyle \sum _j\left[{\displaystyle \underset{s}{\int }\frac{e^{-i{Q}_{\mathrm{e.g}}^{am}{Z}_{mj}\left(x_j,y_j\right)}-1}{-i{Q}_{\mathrm{e.g}}^{am}}{e}^{-i\left(Q_x\cdot x_j+Q_y\cdot y_j\right)}\mathrm{i.e}{x}_j\mathrm{i.e}{y}_j}\right]e^{-i\left(Q_x\cdot X_j+Q_y\cdot Y_j\right)}}\\ ={\displaystyle \sum _j{f}_{mj}\left(Q_{\mathrm{e.g}}^{\mathrm{right}m},Q_x,Q_y\right)e^{-\tilde{i}\left(Q_x\cdot X_j+Q_y\cdot Y_j\right)}}\\ Q_{\mathrm{e.g}}^{am}=\left\{Q_m^{T_m{\tilde{T}}_m},Q_{\mathrm{e.g}}^{R_m{\tilde{T}}_m},Q_{\mathrm{e.g}}^{T_m{\tilde{R}}_m},Q_m^{R_m{\tilde{R}}_m}\right\}\end{array}$$

$$\begin{array}{l}{Q}_{\mathrm{e.g}}^{T_m{\tilde{T}}_m}=k_0\left(n_m+{\zeta }_m\right)\hfill \\ Q_{\mathrm{e.g}}^{R_m{\tilde{T}}_m}=k_0\left(-n_m+{\zeta }_m\right)\hfill \\ Q_{\mathrm{e.g}}^{T_m{\tilde{R}}_m}=k_0\left(n_m-{\zeta }_m\right)\hfill \\ Q_{\mathrm{e.g}}^{R_m{\tilde{R}}_m}=k_0\left(-n_m-{\zeta }_m\right)\hfill \end{array}\}\{\begin{array}{c}{n}_m=\sqrt{n_m-{\text{cos}}^{2}a}\\ {\zeta }_m=\sqrt{n_m-{\text{<figure-callout id="2" label="cos" filenames="DE112010001894B4\_0093.png,DE112010001894B4\_0094.png" state="{{state}}">cos</figure-callout>}}^2\beta }\end{array}$$

It should be noted that these have a large imaginary term in the range of total internal reflection. Here, the integration part of a shape function Zmj (X _j ,Y _j ) in the equation (33) is defined as a shape factor of the unit parts in a repetitive structure as shown in the following formula.
[Formula 34] $$\begin{array}{l}{f}_{mj}\left(Q_{\mathrm{e.g}}^a,Q_{//}\right)\equiv {\displaystyle \underset{s}{\int }\frac{e^{-i{Q}_{\mathrm{e.g}}^a{Z}_{mj}\left(x_j,y_j\right)}-1}{-i{Q}_{\mathrm{e.g}}^a}}{e}^{-i\left(Q_x\cdot x_j+Q_y\cdot y_j\right)}\mathrm{i.e}{x}_j\mathrm{i.e}{y}_j\hfill \\ Q_{//}=\left(Q_x,Q_y\right)\hfill \end{array}$$

$$F_j^{DWBA}\left(\alpha ,\beta ,Q_{//}\right)\equiv {\displaystyle \sum _m{T}_m{\tilde{T}}_m}\left\{\begin{array}{l}{F}_{mj}\left(Q_z^{T_m{\tilde{T}}_m},Q_{//}\right)+R_{\dot{m}}{F}_{mj}\left(Q_z^{R_m{\tilde{T}}_m},Q_{//}\right)\hfill \\ +{\tilde{R}}_{\dot{m}}{F}_{mj}\left(Q_z^{T_m{\tilde{R}}_m},Q_{//}\right)+R_m{\tilde{R}}_{\dot{m}}{F}_{mj}\left(Q_z^{R_m{\tilde{R}}_m},Q_{//}\right)\hfill \end{array}\right\}$$

If the formula (32) is rewritten with this, the following expression follows.
[Formula 35] $$\begin{array}{c}〈{\tilde{\psi }}^f\left(\beta \right)\left|V\right|{\psi }^i\left(a\right)〉={\mathrm{right}}_{c}P{\displaystyle \sum _j{\displaystyle \sum _m{T}_m{\tilde{T}}_m}\left\{\begin{array}{l}{f}_{mj}\left(Q_{\mathrm{e.g}}^{T_m{\tilde{T}}_m}\right)+R_{\dot{m}}{f}_{mj}\left(Q_{\mathrm{e.g}}^{R_m{\tilde{T}}_m}\right)\hfill \\ +{\tilde{R}}_{\dot{m}}{f}_{mj}\left(Q_{\mathrm{e.g}}^{T_m{\tilde{R}}_m}\right)+R_m{\tilde{R}}_{\dot{m}}{f}_{mj}\left(Q_{\mathrm{e.g}}^{R_m{\tilde{R}}_m}\right)\hfill \end{array}\right\}{e}^{-i\left(Q_x\cdot X_j+Q_y\cdot Y_j\right)}}\\ ={\mathrm{right}}_{c}P{\displaystyle \sum _j{f}_j^{DWBA}}\left(a,\beta ,Q_{//}\right)e^{-i\left(Q_x\cdot X_j+Q_y\cdot Y_j\right)}\end{array}$$

$$f_j^{DWBA}\left(a,\beta ,Q_{//}\right)\equiv {\displaystyle \sum _m{T}_m{\tilde{T}}_m}\left\{\begin{array}{l}{f}_{mj}\left(Q_{\mathrm{e.g}}^{T_m{\tilde{T}}_m},Q_{//}\right)+R_{\dot{m}}{f}_{mj}\left(Q_{\mathrm{e.g}}^{R_m{\tilde{T}}_m},Q_{//}\right)\hfill \\ +{\tilde{R}}_{\dot{m}}{f}_{mj}\left(Q_{\mathrm{e.g}}^{T_m{\tilde{R}}_m},Q_{//}\right)+R_m{\tilde{R}}_{\dot{m}}{f}_{mj}\left(Q_{\mathrm{e.g}}^{R_m{\tilde{R}}_m},Q_{//}\right)\hfill \end{array}\right\}$$

Here, even if the microstructure formed on the surface is a single-layer or multi-layer structure, the scattering amplitude can be calculated. The following structure is an example of the microstructure in which a plurality of layers are formed as described above.

11 14 is a cross-sectional view showing a sample model 147 having a structure in which material compositions differ as the height changes. like it in 11 1, in the sample model 147, the material compositions of an end portion 147a of a convex portion and the material compositions of a base portion 147b of the convex portion and a substrate body portion are different from each other. Considering the end portion 147a as a layer L1, and the base portion 147b as a layer L2, the calculation becomes easy. 12 14 is a cross-sectional view showing a sample model 148 having a step in the convex portion. like it in 12 1, in the sample model 148, the step exists between a convex portion end portion 148a and a convex portion base portion 158b. this Additionally, portion 148a can be considered layer L1 and base portion 148b can be considered layer L2. Furthermore 13 14 is a cross-sectional view showing a sample model 149 in which a cap layer 149a is newly formed on a microstructure 149b. In this case, the capping layer 149a can be considered as layer L1 and the microstructure 149b can be considered as layer L2. In such a case, a multilayer structural model is effective.

• Mittlere Position X _j = (X _j, Y _j)
• Versetzung u(X_j)= (u_x(X_j, Y_j), u_y(X_j, Y_j))

When there is a fluctuation of the periodic structure, when a sum with respect to j is obtained, the position coordinate for each cell is divided into an average position and an offset resulting from the fluctuation as shown in the following equation, and thus becomes a scattering cross-section (scattering intensity) is calculated.
[Formula 36]

• middle position X _j = ( X _j , Y _j )
• Displacement u(X _j )= (u _x (X _j , Y _j ), u _y (X _j , Y _j ))

$$I\left(\alpha ,\beta ,Q_{//}\right)={\left(r_{c}P\right)}^{2}N\left\{\langle {\left|F^{DWBA}\right|}^2\rangle +{\left|\langle F^{DWBA}\rangle \right|}^2{e}^{-\frac{Q_u^2{\mathrm{\Delta }}_u^{-2}}{2}}{\displaystyle \sum _{\left|k\right|\ge 1}{e}^{-i{Q}_{//}\left({\overline{X}}_j-{\overline{X}}_{j+k}\right)}}\right\}$$

$$I\left(\alpha ,\beta ,Q_{//}\right)={\left(r_{c}P\right)}^{2}N\left\{\langle {\left|F^{DWBA}\right|}^2\rangle +{\left|\langle F^{DWBA}\rangle \right|}^2{\displaystyle \sum _{\left|k\right|\ge 1}{e}^{i{Q}_{//}\left({\overline{X}}_j-{\overline{X}}_{j+k}\right)}{e}^{-\frac{Q_u^2}{2}{a}^2{\left(1-\text{cos}\left(2\pi \frac{\left|\mathrm{\Delta }{X}_k\right|}{p}\right)\right)}^2}}\right\}$$

〈FDWBA〉

Mittelwert von F^DWBA bezüglich Abweichungen zwischen den Struktureinheiten j

〈|FDWBA|2〉

Mittelwert von |F^DWBA |² bezüglich Abweichungen zwischen den Struktureinheiten j

In this case, equations (12'), (16'), (17'), and the like can be applied without further processing.
[Formula 37] $$I\left(a,\beta ,Q_{//}\right)={\left({\mathrm{right}}_{c}P\right)}^{2}N\left\{〈{\left|f^{DWBA}\right|}^2〉+{\left|〈f^{DWBA}〉\right|}^2{\displaystyle \sum _{\left|k\right|\ge 1}{〈e^{-i{Q}_{//}\left(\mathrm{and}\left(X_j\right)-\mathrm{and}\left(X_k\right)\right)}〉}_j{e}^{i{Q}_{//}\left({\overline{X}}_j-{\overline{X}}_{j+k}\right)}}\right\}$$

$$I\left(a,\beta ,Q_{//}\right)={\left({\mathrm{right}}_{c}P\right)}^{2}N\left\{〈{\left|f^{DWBA}\right|}^2〉+{\left|〈f^{DWBA}〉\right|}^2{e}^{-\frac{Q_{\mathrm{and}}^2{\mathrm{\Delta }}_{\mathrm{and}}^{-2}}{2}}{\displaystyle \sum _{\left|k\right|\ge 1}{e}^{-i{Q}_{//}\left({\overline{X}}_j-{\overline{X}}_{j+k}\right)}}\right\}$$

$$I\left(a,\beta ,Q_{//}\right)={\left({\mathrm{right}}_{c}P\right)}^{2}N\left\{〈{\left|f^{DWBA}\right|}^2〉+{\left|〈f^{DWBA}〉\right|}^2{\displaystyle \sum _{\left|k\right|\ge 1}{e}^{i{Q}_{//}\left({\overline{X}}_j-{\overline{X}}_{j+k}\right)}{e}^{-\frac{Q_{\mathrm{and}}^2}{2}{a}^2{\left(1-\text{cos}\left(2\pi \frac{\left|\mathrm{\Delta }{X}_k\right|}{p}\right)\right)}^2}}\right\}$$

〈FDWBA〉

Mean value of F ^DWBA regarding deviations between the structural units j

〈|FDWBA|2〉

Mean of |F ^DWBA | ² regarding deviations between the structural units j

14 13 is a diagram showing an output of the scattering intensity from a position of a calculated periodic sum when the period of the noise is p=2 in the example of the equation (40). Peaks appearing in positions where the values of the scattering vector Q are multiples of 0.1 result from the original periodic structure. Slightly weaker peaks appearing in the intermediate positions thereof are due to the periodic disturbance.

The specific calculation of the form factor F ^DWBA corresponding to the unit structures that is developed is described below. Here, the term "designed" refers to the fact that, as shown in Equation (36), reflection and diffraction on the surface is taken into account based on DWBA; it should be noted that the form factor F ^DWBA is a function that depends not only directly on the scattering vector Q but also on the angle of incidence α and the angle of exit β. Since the elements of Equation (36) are given by Equation (35), some specific examples will first be determined.

(Cylindrical shape (height H and radius A))

Here, since the height of the cylinder is constant H within the radius A, the following analytical solution can be obtained.
[Formula 38] $$\begin{array}{c}{f}_j\left(Q_{\mathrm{e.g}}^a,Q_{//}\right)={\displaystyle \underset{\mathrm{right}<a}{\int }\frac{e^{-i{Q}_{\mathrm{e.g}}^{a}H}-1}{-i{Q}_{\mathrm{e.g}}^a}{e}^{-i\left(Q_x\cdot x_j+Q_y\cdot y_j\right)}\mathrm{i.e}{x}_j\mathrm{i.e}{y}_j=\frac{e^{-i}{Q}_{\mathrm{e.g}}^{a}H-1}{-i{Q}_{\mathrm{e.g}}^a}}{\displaystyle \underset{0}{\overset{2\pi }{\int }}\mathrm{i.e}\theta {\displaystyle \underset{0}{\overset{a}{\int }}\mathrm{right}\mathrm{i.e}\mathrm{right}{e}^{-i{Q}_{//}\cdot \mathrm{right}\text{cos}\theta }}}\\ =2\pi a\frac{e^{-i{Q}_{\mathrm{e.g}}^{a}H}-1}{-i{Q}_{\mathrm{e.g}}^a}\frac{J\left(a\cdot Q_{//}\right)}{Q_{//}}\left(Q_{//}=\left|Q_{//}\right|\right)\end{array}$$

(One-dimensional trapezoidal lattice)

wobei Parameter des Trapezoids die Länge Wt der oberen Seite, die Länge Wb der unteren Seite und die Höhe H sind.Next, consider a one-dimensional lattice that has an infinitely long trapezoidal shape in the x-direction. Here the integration can also be performed analytically and the following form factor can be obtained.
[Formula 39] $$\begin{array}{c}{f}_j\left(Q_{\mathrm{e.g}}^a,Q_{//}\right)={\displaystyle \underset{-\frac{L_Y}{2}}{\overset{\frac{L_Y}{2}}{\int }}\frac{e^{-i{Q}_{\mathrm{e.g}}^{a}Z\left(y_j\right)}-1}{-i{Q}_{\mathrm{e.g}}^a}}{e}^{-i{Q}_y\cdot y_j}\mathrm{i.e}{y}_j{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{e}^{-i{Q}_x\cdot x}}\mathrm{i.e}x\\ =2\pi \delta \left(Q_x\right)\left[e^{-i\frac{W_b}{2}}{Q}_y\frac{e^{-i\left(Q_{\mathrm{e.g}}^a-\frac{W_b-W_t}{2H}{Q}_y\right)H}-1}{-i\left(Q_{\mathrm{e.g}}^a-\frac{W_b-W_t}{2H}{Q}_y\right)}-e^{-i\frac{W_b}{2}{Q}_y}\frac{e^{-i\left(Q_{\mathrm{e.g}}^a+\frac{W_b-W_t}{2H}{Q}_y\right)H}-1}{-i\left(Q_{\mathrm{e.g}}^a+\frac{W_b-W_t}{2H}{Q}_y\right)}\right]\end{array}$$

where parameters of the trapezoid are the top side length Wt, the bottom side length Wb and the height H.

(shape factor in the case of a complicated shape function)

$$\begin{array}{c}{\displaystyle \underset{-\frac{L_Y}{2}}{\overset{\frac{L_Y}{2}}{\int }}\frac{e^{-i{Q}_z^a{Z}_j\left(y_j\right)}-1}{-i{Q}_z^a}}{e}^{-i{Q}_y\cdot y_j}d{y}_j\Rightarrow {\displaystyle \underset{s=\frac{n-1}{2}}{\overset{s=\frac{n-1}{2}}{\int }}\frac{e^{-i{Q}_z^a{Z}_j\left(\mathrm{\Delta }y\cdot s\right)}-1}{-i{Q}_z^a}}{e}^{-i\left(\mathrm{\Delta }{Q}_y\cdot h\right)\left(\Delta y\cdot s\right)}\mathrm{\Delta }y\\ F_j\left(Q_z^a\cdot h\right)=\mathrm{\Delta }y{\displaystyle \underset{s=\frac{n-1}{2}}{\overset{s=\frac{n-1}{2}}{\int }}\frac{e^{-i{Q}_z^a{Z}_j\left(\mathrm{\Delta }y\cdot s\right)}-1}{-i{Q}_z^a}}{e}^{-2\pi \frac{h\cdot s}{n}}\left(\mathrm{\Delta }y=\frac{L_Y}{2},\mathrm{\Delta }{Q}_y=\frac{2\pi }{L_Y}\right)\\ \left(n,s,h:\text{Integer}\right)\end{array}$$

The above two examples are cases where the integration can be performed analytically. However, performing such an integration is generally not easy. Therefore, a method that can be applied to a complicated shape is considered. For example, since in a one-dimensional lattice having the structure in which the gate structure of an LSI is modeled and the in 9 as shown, it is impossible to perform analytical integration, it is necessary to calculate the form factor of equation (35) by discrete numerical integration and determine the scattering intensity. Here, in view of the actual execution, it is necessary to perform the integration as effectively as possible. Before the calculation is specifically carried out, the features of equations (38) to (40) will be described. In any case, the diffraction peaks appear, which are due to the periodic structure contained in 14 shown, and the X-ray scattering intensity reflecting the surface nanostructure is only observed in the diffraction peaks. Consequently, the calculation of the scattering intensity is preferably carried out only in such a diffraction angle in which the scattering vector Q// parallel to the surface satisfies the scattering conditions 2Lsinθ=hλ. In this case, as shown below, it is possible to effectively use Fast Fourier Transform (FFT), which is the Fourier transform of a periodic structure.
[Formula 40] $$\begin{array}{c}{f}_j\left(Q_{\mathrm{e.g}}^a,Q_x,Q_y\right)={\displaystyle \underset{-\frac{L_Y}{2}}{\overset{\frac{L_Y}{2}}{\int }}\frac{e^{-i{Q}_{\mathrm{e.g}}^a{Z}_j\left(y_j\right)}-1}{-i{Q}_{\mathrm{e.g}}^a}}{e}^{-i{Q}_y\cdot y_j}\mathrm{i.e}y{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{e}^{-i{Q}_x\cdot x}}\mathrm{i.e}x\\ =2\pi \delta \left(Q_x\right){\displaystyle \underset{-\frac{L_Y}{2}}{\overset{\frac{L_Y}{2}}{\int }}\frac{e^{-i{Q}_{\mathrm{e.g}}^a{Z}_j\left(y_j\right)}-1}{-i{Q}_{\mathrm{e.g}}^a}}{e}^{-i{Q}_y\cdot y_j}\mathrm{i.e}{y}_j\end{array}$$

$$\begin{array}{c}{\displaystyle \underset{-\frac{L_Y}{2}}{\overset{\frac{L_Y}{2}}{\int }}\frac{e^{-i{Q}_{\mathrm{e.g}}^a{Z}_j\left(y_j\right)}-1}{-i{Q}_{\mathrm{e.g}}^a}}{e}^{-i{Q}_y\cdot y_j}\mathrm{i.e}{y}_j\Rightarrow {\displaystyle \underset{s=\frac{n-1}{2}}{\overset{s=\frac{n-1}{2}}{\int }}\frac{e^{-i{Q}_{\mathrm{e.g}}^a{Z}_j\left(\mathrm{\Delta }y\cdot s\right)}-1}{-i{Q}_{\mathrm{e.g}}^a}}{e}^{-i\left(\mathrm{\Delta }{Q}_y\cdot H\right)\left(\Delta y\cdot s\right)}\mathrm{\Delta }y\\ f_j\left(Q_{\mathrm{e.g}}^a\cdot H\right)=\mathrm{\Delta }y{\displaystyle \underset{s=\frac{n-1}{2}}{\overset{s=\frac{n-1}{2}}{\int }}\frac{e^{-i{Q}_{\mathrm{e.g}}^a{Z}_j\left(\mathrm{\Delta }y\cdot s\right)}-1}{-i{Q}_{\mathrm{e.g}}^a}}{e}^{-2\pi \frac{H\cdot s}{n}}\left(\mathrm{\Delta }y=\frac{{\mathrm{<figure-callout\; id="2"\; label="L\; Y"\; filenames="DE112010001894B4\_0093.png,DE112010001894B4\_0094.png"\; state="\{\{state\}\}">L</figure-callout>}}_Y}{2},\mathrm{\Delta }{Q}_y=\frac{2\pi }{L_Y}\right)\\ \left(n,s,H:\text{integer}\right)\end{array}$$

$$\mathrm{\Delta }{Q}_y\cdot h=2\pi /L_Y\cdot h$$

Here, it was found that from Equation (45) showing the diffraction conditions and the definition, Equation (46) gives the peak positions of diffraction.
[Formula 41] $$Q_y=4\pi \text{sin}\theta /\lambda$$

$$\mathrm{\Delta }{Q}_y\cdot H=2\pi /L_Y\cdot H$$

In particular, given a cross-sectional shape function Z(y) given by FFT shown in equation (44), the scattering intensity in all diffraction peak positions for the values of Q _z ^a can be calculated simultaneously. Although the one-dimensional lattice endlessly continuous in the x-direction has been considered in the above description, when this is expanded into a two-dimensional structure Z(x,y), the equation below follows.
[Formula 42] $$\begin{array}{c}{f}_j\left(Q_{\mathrm{e.g}}^a,Q_x,Q_y\right)={\displaystyle \underset{-\frac{L_Y}{2}}{\overset{\frac{L_Y}{2}}{\int }}{\displaystyle \underset{-\frac{L_X}{2}}{\overset{\frac{L_X}{2}}{\int }}\frac{e^{-i{Q}_{\mathrm{e.g}}^a{Z}_j\left(x_j,y_j\right)}-1}{-i{Q}_{\mathrm{e.g}}^a}}{e}^{-i\left(Q_x\cdot x_j+Q_y\cdot y_j\right)}\mathrm{i.e}{x}_j\mathrm{i.e}{y}_j}\\ \Rightarrow {\displaystyle \sum _{t=\frac{m-1}{2}}^{t=\frac{m-1}{2}}{\displaystyle \sum _{s=\frac{n-1}{2}}^{s=\frac{n-1}{2}}\frac{e^{-i{Q}_{\mathrm{e.g}}^a{Z}_j\left(\mathrm{\Delta }x\cdot s\mathrm{\Delta }y\cdot t\right)}-1}{-i{Q}_{\mathrm{e.g}}^a}{e}^{-i\left(\mathrm{\Delta }{Q}_x\cdot H\right)\cdot \left(\mathrm{\Delta }x\cdot s\right)}{e}^{-i\left(\mathrm{\Delta }{Q}_y\cdot k\right)\cdot \left(\mathrm{\Delta }y\cdot t\right)}\mathrm{\Delta }x\mathrm{\Delta }y}}\\ f_j\left(Q_{\mathrm{e.g}}^a,H,k\right)=\mathrm{\Delta }x\mathrm{\Delta }y{\displaystyle \sum _{t=\frac{m-1}{2}}^{t=\frac{m-1}{2}}{\displaystyle \sum _{s=\frac{n-1}{2}}^{s=\frac{n-1}{2}}\frac{e^{-i{Q}_{\mathrm{e.g}}^a{Z}_j\left(\mathrm{\Delta }x\cdot s\mathrm{\Delta }y\cdot t\right)}-1}{-i{Q}_{\mathrm{e.g}}^a}{e}^{-2\pi i\frac{H\cdot s}{n}}{e}^{-2\pi i\frac{k\cdot t}{m}}}}\\ \left(\mathrm{\Delta }x=\frac{L_X}{n},\mathrm{\Delta }{Q}_x=\frac{2\mathrm{\pi }}{L_X},\mathrm{\Delta }y=\frac{L_Y}{m},\mathrm{\Delta }{Q}_y=\frac{2\pi }{L_Y},n,m,s,t,H,k:\text{integer}\right)\end{array}$$

An example of calculating the scattering intensity by using the calculation method with Equation (47) when a complicated cross-sectional shape is provided in which the analytic integration shown in 9 shown cannot be executed. Here, considering not only the shape of the trapezoid but also the roundness of each edge, what effects are exerted on the scattering pattern is evaluated. The 15 and 17 12 are cross-sectional views of sample models having a line segment whose cross section is trapezoidal. Compared to the model that is in 15 shown is in the model presented in 17 as shown, the shape of the base portion is significantly modified. The 16 and 18 are corresponding diagrams showing the results of simulations performed on the models presented in 15 and 17 are shown have been carried out. It can be seen that the difference in cross-sectional shape causes a clear difference in the scattering pattern.

(Sample model in which a surface layer is covered with a single layer or a plurality of layers)

The method of using the structure factor in the complicated shape function described above can also be applied to a sample model in which a surface layer is covered with a single layer or a plurality of layers (films). 19 is a cross sectional view of a sample model, which has a line segment, the surface of which is covered with a layer. 20 Fig. 12 is a diagram showing the result of a simulation performed on the model of 19 was executed. like it in 20 is shown, in such a sample model, a core portion (a first substantial area) in which trapezoids are formed uniformly in the parallel x-direction and one or more layer portions (a second substantial area) formed in the form of layers on the core portion are formed as the unit structure on the sample surface. The material compositions of the layer sections are different from those of the core section. The layered portions may be formed with a plurality of films. It is possible to calculate the X-ray scattering intensity on such a sample model.

In the manufacturing process of various devices, when inspecting a member formed by coating an interface layer or a metal layer on a line and space structure, a thin film is preferably formed on a line, a side wall and a bottom portion so that the thickness of the film is as constant as possible. In this case, with the sample model described above, it is possible to carry out non-destructive measurement for determining whether the film is formed uniformly on the line, the side wall and the bottom portion.

(Sample model having a line section where an asymmetric side wall is formed)

The method of using the structure factor in the complicated shape function can also be applied to a sample model having a line portion in which an asymmetric side wall is formed. 21 Fig. 14 is a cross-sectional view of the sample model having the line portion in which the asymmetric side wall is formed. 22 Fig. 12 is a diagram showing the result of a simulation performed on the model of 21 was executed. like it in 21 shown, even when the sample model has the unit structure in which a cross-sectional shape perpendicular to the x-direction parallel to the sample surface is formed into asymmetric trapezoids, it is possible to calculate the X-ray scattering intensity. In the above example, a line and space structure is formed uniformly in the x-direction parallel to the sample surface.

When lines are formed with a resist, the other portions are etched away, and thus a line and space structure is formed, and an asymmetric sidewall structure can be formed by anisotropic exposure to an etching gas or the like. Even in this case, with the sample model described above, it is possible to have sufficient detection sensitivity to the asymmetry of the sidewall angle. Consequently, it is possible to use this as an observation of a process in which asymmetry is expected to play a role.

examples

[Results of an experiment]

An experiment was conducted by using a sample in which the unit structures have a repetitive periodic structure. A sample was used as the sample, the microstructure of which was determined to be accurate, ie with which a calibration is carried out. 23 and 24 are an SEM photo from above and a TEM photo in cross section. Like it in the 23 and 24 1, on the surface of the sample, line portions 146a having uniform cross-sectional shapes in the x-direction are formed as repeating units so that they are uniformly spaced in the y-direction. That is, the cross-sectional shape in the yz plane is analyzed. Further, space portions 146b are formed between the line portions 146a. It can be found that the shape whose cross section is trapezoidal is not simply trapezoidal but has round portions with a certain radius of curvature in a convex end portion 146c of an upper side and a concave base portion 146d.

25 Fig. 12 is a diagram showing a cross section of a sample model of the sample described above. The diagram is presented in such a way that in addition to the characteristics of the trapezoid, such as the length of the upper side, the length of the lower side and the height, parameters are introduced for the radius of curvature of the convex end portion of the upper side and the radius of curvature of the concave base portion, and that the radius of curvature of the round sections described above can be reproduced.

The X-ray scattering intensity was actually measured on the above sample, the X-ray scattering intensity was calculated for the above sample model, and thus the optimal values for the parameters were obtained by means of a fit. 26 Fig. 12 is a graph showing actually measured X-ray intensity. In 26 portions showing high X-ray scattering intensity are shown in white. On the other hand 27 a graph showing an X-ray scattering intensity calculated by simulation using the optimal parameters. Like it in the 26 and 27 is shown, it can be seen that the scattering intensity is approximately the same.

100

Oberflächenmikrostruktur-Messsystem

110

Röntgenstreuungs-Messsystem

113

Monochromator

114 und 115

Kollimationsblock

115a

Probenhalter

116

zweidimensionaler Detektor

117

Strahlstopper

118

Schlitz

119

scharfe Kante

120

Oberflächenmikrostruktur-Analyseeinrichtung

121

Parameterbeschaffungsabschnitt

122

Formelspeicherabschnitt

123

Simulationsabschnitt

124

Fitabschnitt

125

Ausgabeabschnitt

140 und 145

Probe

141 und 146

Probenoberfläche

146a

Linienabschnitte

146b

Raumabschnitt

146c

konvexer Endabschnitt

146d

Basisabschnitt

147, 148 und 149

Probenmodel

147a und 148a

Endabschnitt

147b und 148b

Basisabschnitt

149a

Abdeckschicht

149b

Mikrostruktur

The 28 and 29 are respective graphs showing the actually measured X-ray scattering intensity and the calculated X-ray scattering intensity at the first and third peaks with respect to the above results. The figures show that the actually measured values and the calculated values are essentially the same. As described above, it was found that the cross-sectional shape in the yz plane obtained by determining the parameters corresponds substantially to the cross-section of the TEM photograph shown in FIG 24 , and it has been found that the method of measuring the surface microstructure according to the present invention is effective.

100

Surface Microstructure Measurement System

110

X-ray scattering measurement system

113

monochromator

114 and 115

collimation block

115a

sample holder

116

two-dimensional detector

117

beam stopper

118

slot

119

sharp edge

120

Surface microstructure analyzer

121

parameter acquisition section

122

formula storage section

123

simulation section

124

fit section

125

output section

140 and 145

sample

141 and 146

sample surface

146a

line sections

146b

space section

146c

convex end section

146d

base section

147, 148 and 149

sample model

147a and 148a

end section

147b and 148b

base section

149a

covering layer

149b

microstructure

Claims (13)

A surface microstructure measuring method for measuring a microstructure (149b) of a sample surface (141, 146), the method comprising the steps of: irradiating the sample surface (141, 146) with X-rays at a grazing incidence angle and measuring a scattering intensity; Assuming a sample model having a microstructure (149b) on a surface (141, 146) in which one or more layers are formed in a direction perpendicular to the surface (141, 146) and unit structures are formed periodically in a direction parallel to the surface (141, 146 ) are arranged within the layers, calculating a scattering intensity of X-rays scattered by the microstructure (149b) based on the refractive indices of the respective layers of the one or more layers in a direction parallel to the surface (141, 146) within the layers, and performing a fit of the scattering intensity of the X-ray radiation calculated from the sample model to the measured scattering intensity; and determining, as a result of the fitting, an optimal value of a parameter for specifying a shape of the unit structures, in which assuming that the unit structures have fluctuations in positions relative to an exact periodic position and the fluctuations in positions only from a relative positional relation between the unit structures, the scattering intensity of the X-rays is calculated in which the unit structures are formed by a uniform substantial area and a uniform void area in the layers, and the scattering intensity of the X-rays caused by the uniform substantial area is calculated. Surface microstructure measurement method claim 1 , in which, assuming that the unit structures have fluctuations in positions relative to an exact periodic position and the fluctuations in positions do not depend on the difference between respective positions and are random, the X-ray scattering intensity is calculated. Surface microstructure measuring method according to any one of the preceding claims, wherein when the fluctuations in the positions of the unit structures have a periodicity, using an amplitude and a period of the fluctuations in the positions to express a mean square of the fluctuations in the positions of the unit structures, the scattering intensity of the X-ray is calculated becomes. A surface microstructure measuring method according to any one of the preceding claims, wherein in a sample model in which the unit structures have the substantial area within a cylinder, the scattering intensity of the X-ray is calculated. Surface microstructure measurement method according to one of Claims 1 until 3 wherein in a sample model in which the unit structures have the substantial area within a trapezoid uniform in an x-direction parallel to the sample surface (141, 146), the scattering intensity of the X-ray is calculated. Surface microstructure measurement method according to one of Claims 1 until 3 , wherein in a sample model in which the unit structures have the uniform substantial area in an x-direction parallel to the sample surface (141, 146) and in a y-direction parallel to the sample surface (141, 146) and perpendicular to the x-direction in elements are divided, an integral is approximated by a sum of the elements, and thus the scattering intensity of X-ray scattering is calculated. Surface microstructure measurement method claim 6 , in which a sample model in which the unit structures have a substantial area having a uniform cross-sectional structure in the x-direction is assumed, either a radius of curvature of a convex end portion of both ends of an upper side or a radius of curvature of a concave base portion of both ends of a lower one Side of the cross-sectional shape is contained in a parameter and thus the scattering intensity of the X-ray radiation is calculated. Surface microstructure measurement method claim 6 , wherein in a sample model in which the unit structures have a first substantial region formed in a shape of a trapezoid uniform in the x-direction and one or more second substantial regions whose element of material differs from a element of material of the first substantial region, and which are formed in layers on the first substantial region, the scattering intensity of the X-ray is calculated. Surface microstructure measurement method claim 6 in which, in a sample model in which the unit structures have the substantial region uniform in the x-direction and in which a cross-sectional shape perpendicular to the x-direction is asymmetrically trapezoidal, the X-ray scattering intensity is calculated. Surface microstructure measurement method according to one of Claims 1 until 3 , in which in a sample model in which the unit structures have a substantial region that has a periodic structure in both an x-direction parallel to the sample surface (141, 146) and a y-direction parallel to the sample surface (141, 146) and perpendicular to the x-direction and divided into elements in the x- and y-directions parallel to the sample surface (141, 146), the scattering intensity of the X-ray from each of the elements is integrated by a sum of the elements. Surface microstructure measuring method according to any one of the preceding claims, wherein the scattering intensity of the X-rays scattered by the microstructure (149b) is calculated taking into account the effects of diffraction and reflection produced by several different layers in the sample model. A surface microstructure measurement data analysis method for measuring a microstructure (149b) on a sample surface (141, 146), the method causing a computer to perform the steps of: assuming a sample model having a microstructure (149b) on a surface (141, 146) in which one or more layers are formed in a direction perpendicular to the surface (141, 146) and unit structures are formed in a direction parallel to the surface (141, 146) are arranged periodically within the layers, calculating a scattering intensity of X-rays scattered by the microstructure (149b), taking into account diffraction and reflection effects generated by the layers, based on the refractive indices of the respective layers of the one or more layers in the direction perpendicular to the surface (141, 146), and fitting the scattering intensity of the X-rays calculated from the sample model to a scattering intensity actually measured by irradiating the sample surface (141, 146) with X-rays at a grazing incidence angle; and determining, as a result of the fitting, an optimal value of a parameter for specifying a shape of the unit structures, wherein, assuming that the unit structures have fluctuations in positions relative to an exact periodic position and the fluctuations in positions depend only on a relative positional relation between the unit structures, the scattering intensity of X-rays is calculated, in which the unit structures are formed by a uniform substantial area and a uniform void area in the layers, and the scattering intensity of the X-ray caused by the uniform substantial area is calculated. Surface microstructure measuring system suitable for measuring a microstructure (149b) on a sample surface (141, 146), the device comprising: an X-ray scattering measuring device (110) and a surface microstructure analysis device (120), the X-ray scattering measuring device (110 ) comprising: a monochromator (113) which spectrally reflects X-rays emitted by an X-ray source; a slit portion (118) which limits a spot size of the spectrally reflected X-ray on the sample surface (141, 146) to 30 µm or less; a sample holder (115a) which allows rotation to change both an angle of incidence of the spectrally reflected X-rays on the sample surface (141, 146) and rotation within a plane of the sample surface (141, 146), and which holds the sample (140, 145 ) carries; and a two-dimensional detector (116) that measures the scattering intensity of X-rays scattered on the sample surface (141, 146), wherein the surface microstructure analysis device (120) comprises a computer, wherein a simulation section (123) controlled by the Computer is performed assuming a sample model having a microstructure (149b) on a surface (141, 146) in which one or more layers are formed in a direction perpendicular to the surface (141, 146) and unit structures periodically in a direction parallel to the surface (141, 146) within the layers, a scattering intensity of the X-rays scattered by the microstructure (149b) due to the refractive indices of the respective layers of the one or more layers in the direction perpendicular to surface (141, 146) and a fitting section (124) performed by the computer that fits the X-ray scattering intensity calculated for the sample model to the measured scattering intensity and, as a result of the fit, an optimum value a parameter for specifying a shape of the unit structures, in which, assuming that the unit structures have fluctuations in positions relative to an exact periodic position and the fluctuations in positions depend only on a relative positional relation between the unit structures, the scattering intensity of the X-rays is calculated in which the unit structures are formed by a uniform substantial area and a uniform void area in the layers, and the scattering intensity of the X-ray caused by the uniform substantial area is calculated.

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